LIBRARY OF CONGRESS. 



she: TJ S'A 

UNITED STATES OF AMERICA. 









SLIDE VALVE GEARS. 

A 

NEW GRAPHICAL METHOD 

FOR 

ANALYZING THE ACTION OF SLIDE VALVES, 

MOVED BY 

ECCENTRICS, LINK-MOTIONS AND CUT-OFF GEARS, 

*Ojferi?tg easy means for properly designing Valves and 

Valve- Gears, and for establishing the cojnparative 

merits of their various co?zstructions, 

BY 

HUGO BILGRAM, M. & 

— ^ Mo. SCGo b 



PHILADELPHIA : 
CLAXTON, REMSEN & HAFFELFINGER, 

Nos. 624^626 &. 628 Market Street. 
1878. 



3ss 



Entered, according to Act of Congress, in the year 1877, by 

HUGO BILGRAM, 

in the Office of the Librarian of Congress at Washington. 





Lehman & Btffton, Printers, 
418-422 Library St., Philada. 



Ill 



PREFACE. 

The most successful method of studying and analyzing the 
action of the slide valve, in its various modes of application, 
is that which is based on the use of graphical diagrams, among 
which the one discovered by Zeuner, * is pretty generally con- 
sidered the best. 

This diagram shows at a glance what, by the aid of work- 
ing models, can be seen only by carefully watching the move- 
ment of the slide valve in relation to that of the piston ; it 
shows how an alteration of any of the essential members of 
the valve gear affects the working of the valve ; it is admi- 
rably adapted to draw a comparison between different valve 
gears and link motions on their intrinsical as well as their 
relative merits ; it also furnishes the means, by a reversed 
-application, of finding the proper proportions of valves and 
the valve -moving devices when a given effect is desired, and 
how any valve gear can be proportioned to the best advan- 
tage. In short, it meets all the principal requirements of the 
case. Only one feature is in some respects adverse to a general 
use of this graphical method ; in some instances its applica- 
tion requires algebraical calculations, or else an intricate 
geometrical procedure, and for this reason it can be used only 
by those engineers who have enjoyed a theoretical education. 

In his efforts to fully open this field to every intelligent 
mechanic, the author found that a modification of the Zeuner 



*Schiebersteuerungen, by Prof. Dr. Gustav Zeuner 



IV 

Diagram renders the subject in all its bearings purely graphi- 
cal, and no knowledge of mathematics beyond the most ele- 
mentary propositions of geometry is required to understand 
the demonstrations as well as the applications of this new 
diagram. Nor is this the only advantage gained over the 
original form ; for by this modification the diagram profited 
in several other respects, especially in regard to clearness, 
and will therefore give better satisfaction, even to engineers 
familiar with mathematics. 

The following discussions will be as brief as is consistent with 
a thorough and fundamental treatment ; and the leading propo- 
sitions will be amply illustrated. 



V 
CONTENTS. 

. Mm*. ■ 

INTRODUCTION: paob 

PeOPEE DlSTEIBUTION OF STEAM, 1 

PART I. 

The Common Slide Valve, 7 

The Valve Diagram,..* •> 

The Angle of Advance, IS 

Width of Ports, 20 

Velocity of the Valve, 22 

Practical use of the Diagram (Problems), 25 

The Scope of the Common Slide Valve, 32 

Irregularities of the Crank Motion (Problems), 34 

PART II. 

Link Motions, 45 

The Stephenson Line: Motion, 46 

The Theory, 49 

The Valve Diagram, , 56 

The\Link^and its Suspension, 60 

Equalization of the Lead, 62' 

Problems, 65 

Irregularities and their Rectification, 69 

The Gooch Link Motion, 78- 

PART III. 

Independent Cut-off Gears, 87 

The Gonzenbach^ Cut-off Geae, 91 

The Meyer Cut-off, Geae, 97 

The Link Expansion Geae, 107 

The Bilgeam Cut-off Geae ; 119 



INTRODUCTION. 



Proper Distribution of Steam. 

The steam engine is a mechanical contrivance for convert- 
ing the power, developed by the evaporation of water, into 
such a form that it can be utilized for driving machinery or 
for other industrial purposes. After its generation in the 
boiler, the steam is caused to exert its pressure alternately 
upon both sides of the piston, imparting a reciprocating mo- 
tion, which is converted into a rotary one by a connecting-rod 
and crank. The valves admitting the steam alternately to 
both ends of the cylinder, and opening an exit for the steam 
after the performance of its duty, are named" distribution- 
valves " and the mechanism operating them is termed ''valve- 
gear. ' ' 

We shall now consider how the admission of steam should be 
regulated in order that the greatest amount of useful effect 
will be realized ; this will enable us to judge how near the 
valve-gears to be considered hereafter approximate to perfec- 
tion. 

I. Expansion. To the engineer, the most valuable proper- 
ty of steam is its elasticity, which can most effectively be 
utilized by admitting steam to the cylinder for a portion of 
the stroke only. The production of work per stroke will then 
comprise two stages : (1) The direct action of steam, for the 
time of admission, when the steam exerts its initial pressure 
upon the piston, and (2) The action of the expanding steam ; 
in which period the pressure is gradually diminishing. 

For economy of steam it is most advantageous to expand 



until the terminal pressure upon the piston equals the average 
Motional resistance of the engine. In practice, however, the 
expansion is not always carried thus far, as an engine accom- 
plishing this result generally demands a higher price than 
one with a single slide-valve, and purchasers of engines fre- 
quently prefer to save in the first cost. 

When the load on the engine is subject to variation, the 
admission of steam must be regulated accordingly. The effec 
tive steam pressure upon the piston is usually more or less 
reduced by throttling the admitted steam by a valve that is 
operated by the governor. A more economical plan, however, 
is to vary the degree of expansion by cutting the steam-off 
at an earlier or later stage of the stroke, if the valve-gear 
admits of such a variation. 

II, Compression. That end of the cylinder towards which 
the piston is moving must be in communication with the ex- 
haust passage. In condensing engines the exhaust passage is 
in communication with the condenser, where a partial vacuum 
is produced by the condensation of the exhaust steam. Other 
engines exhaust into the air, and have therefore a back press- 
ure equal to, or slightly above, that of the atmosphere. 

It is of advantage to interrupt the exhaust a short time 
before the termination of the stroke, and to compress the thus 
insulated exhaust steam by the piston. At the first glance 
this statement may appear absurd ; but it is nevertheless con- 
sistent with both theory and practical experience, for the fol- 
lowing reasons : 

The " clearance/ ' — which is the space left at the end of 
the stroke, between the piston and the end of the cylinder, 
together with the space of the steam passage between the 
valve and the cylinder, — is alternately brought in communi- 
cation with the steam chest and with the exhaust passage, and 
is thus filled with steam and emptied again. The direct 



action of this steam is evidently lost, and only part of the ex- 
pansive power of the same is made available. This continual 
loss of power can, however, be moderated by compressing into 
the clearance part of the exhaust steam, raising its pressure to 
that in the boiler. "When thereafter the valve opens the port 
for admission, no steam is required to fill the clearance, and 
the full power of all the admitted steam will be turned into 
useful work. If now the steam is cut off so early in the stroke 
that the terminal pressure will be equal to that in the ex- 
haust passage, the re-expansion of the previously compressed 
steam will augment the production of work just so much as 
to restore the work expended in compression. This shows 
that under the said conditions the clearance will cause no loss 
of steam. It must, however, be remembered that in the first 
place we took no account of friction. The space through 
which the piston travels during compression, is entirely lost 
for the production of work ; therefore it is evident that an 
engine with much clearance and corresponding compression, 
though theoretically equal to one with but little clearance and 
compression, is practically inferior, for since the travel of the 
piston must be greater it will involve greater loss from fric- 
tion. In the second place, we assumed the expansion carried 
to a degree higher than that generally used. It is, therefore, . 
advisable to compress to a less degree, especially if not much, 
expansion is used ; for the work expended for compression 
will be but partially restored by the following re-expansion. 

A second reason for prematurely closing the exhaust is due 
to the reaction of the compressed exhaust steam upon the pis- 
ton, — commonly called "cushioning," — which assists in the 
reversal of the motion of the piston and other reciprocating 
parts of the engine and lessens the wear in bearings, and also 
the thumping caused by lost motion which is usually present 
to some extent. 



III. Lead. Experience has taught that both the admission 
and the exhaust of steam should be commenced a short time 
before the crank passes the centre. Either port will thereby 
be opened to some extent at the beginning of the stroke. The 
width of the opening at that time is termed "lead," while 
the angle through which the crank passes from the moment 
of the opening of the port until it reaches the centre is the 
"lead angle." 

Cne reason is obvious. The steam does not fill the clear- 
ance, nor does the exhaust steam reduce its pressure to that 
of the exhaust passage, the very instant the port for either 
admission or exhaust is opened ; hence the advantage of a 
premature opening on either side. 

As a general rule the lead on the exhaust side is greater 
than that on the steam side, as the former has the function of 
releasing the pressure of a volume of steam filling the entire 
cylinder, while the latter need only fill the clearance with 
live steam. The opening of the exhaust is mostly effected 
w 7 hile the piston is still an appreciable distance from the end 
of its stroke, and is more commonly called "release," while 
the admission mostly begins when the piston has practically 
arrived at the end of its travel. 

The most advantageous amount of lead on the exhaust side 
depends mainly on the terminal pressure of the expanded 
steam ; and should, therefore, be greater on engines with late 
cut-off. High speed engines need more lead on both sides 
than slow running ones, to allow the proper time for the 
functions of lead. Excessive lead on either side always re- 
sults in waste of power. 

Lead is also a very convenient means for supplementing the 
cushioning if compression is insufficient for this purpose ; it can be 
adjusted until the engine ceases to thump. High speed engines 
and beam engines need more lead on this account, than others^ 
because of the greater momentum to be overcome. 



PART I. 

THE COMMON SLIDE VALVE. 



The Common Slide Valve. 
The usual form of the common slide valve is shown in Fig. 
1, where /Sand 8 ' represent the steam passages of the cylinder, 




kick i a&vaf 

Fig. 1. 

and S° the exhaust passage. Tis the valve, the cavity v of 
which effects alternately a connection of the steam ports with 
the exhaust port. The valve is encased in the steam chest, 
into which the live steam is admitted. It is moved by the 
valve stem passing through a stuffing box of the steam chest. 

When the valve is in its neutral position, as shown, its face 
overlaps the steam ports on both sides. The distance a b is 
called the outside or steam side lap, andc d is the inside or ex- 
haust side lap. 

In the following considerations we shall term "positive " 
any movement of the valve from its neutral position towards 
the valve stem, in contradistinction from a movement in the 
opposite or "negative" direction. A positive movement of 
the valve connects the port 8' with the exhaust passage 8° 
through the cavity 0, and opens the port 8 for admission of 
steam from the steam chest. The width of the opening 
of either port depends en the distance of the valve from 



8 

its neutral position, for the time being. This distance, or the 
"movement " of the valve, must exceed the inside lap before 
he port S' will be opened for exhaust ; and it must be greater 
than the outside lap, to effect an opening of the port S for 
admission of live steam. For a given movement of the valve- 
we find the opening of the steam ports for either exhaust or 




Fig. 2. 

admission by deducting from this movement the inside or 
outside lap respectively. When this difference is equal to or 
greater than the width of the port, the latter is fully opened. 
The effect produced by a motion of the valve in the opposite 
(negative) direction is similar, but reversed as regards the pass- 
ages #and S\ 

The valve is moved by means of the eccentric I, (Fig. 2), 
which can properly be considered as a small crank, the pin of 
which is enlarged so much as to assume the form of a disc that 
allows the shaft to pass through it. The eccentric can there- 
fore be represented by the line J, that connects the centre of 
the shaft, O, with the centre of the eccentric disc, I, and 
which is supposed to act like a crank. 



The Valve Diagram. 

We shall first consider horizontal engines of the usual con- 
struction, m which both the connecting rod and the eccentric 
rod, are operated in parallel directions. 

The eccentric of such engines is keyed to the crank shaft so 
that the centre line 01 (Fig. 2) of the eccentric forms an 
obtuse angle with the centre line OK of the crank. As the 
•crank shaft rotates, the end P of the eccentrie rod, being 
.guided in a right line by the valve stem, receives a recipro- 
cating motion while the rod itself is caused to oscillate. The re- 
ciprocating motion of the end P is transmitted by means of the 




i / 
» / 



valve stem to the slide valve. The oscillating movement in- 
troduces a very intricate factor, since it affects the horizontal 
-extension of the eccentric rod. We shall, however, at present 



10 

leave this factor altogether out of account, and shall examine 
it later. 

It is necessary to locate the centre of vibration of the endP 
of the eccentric rod on account of its corresponding relation 
to the neutral position of the valve. The vibration of this 
point being a direct result of the circular movement of the 
other end, I, of the eccentric rod, it is evident that the centres 
of both movements must correspond. Therefore, if the end I 
of the rod IP be transposed to the centre of its motion, (which 
is the centre of the 'crank shaft) its other end P will occupy 
the centre of its' vibration. It will now be understood that 
the movement of the pin P depends on the horizontal move- 
ment of the centre, 7, of the eccentric ; and can therefore be 
measured by the horizontal distance of the point /from the 
vertical centre line Y. 

When the crank is on its centre OK (Fig. 3) and J is the 
corresponding position of the eccentric, the distance Iy will 
represent the valve movement. 

As the crank moves from JTto IP through the angle a, the 
eccentric will move through the same angle to T, and Ty y 
will now be the distance of the valve from its neutral posi- 
tion. 

Let us now mark the point Q in the circle Z representing- 
the orbit of the eccentric ; making the angle XOQ equal to 
the angle 10 Y (lioth of which are marked 6 in Fig. 3), and 
draw the line Q k perpendicular to the extension of the line 
OK of the crank. The triangles QOk and T 0y y will then be 
equal, since the angles QOk and Q JcO are equal respectively 
to the angles T Oy' and Vy' 0, and the side OQ equals the 
side 01 ; hence the line Qk equals l'y y and therefore is equal 
to the movement of the valve for the crank angle a. The 
same relation can be show T n, in the same manner, for any 
other crank angle, and the interesting fact is thcreb/ demon- 
strated that, for any angle of the crank, the distance of the 



11 

point Qfrom the centre line of the crank is equal to the corre- 
sponding movement of the slide valve. 

To find the openings of the ports for either admission 
or exhaust, the inside or outside lap is to be deducted from 
this movement. This subtraction can be graphically per- 
formed by sweeping two circles, L and I, fromQ with 
radii equal to the outside and inside laps, as shown in 
Fig. 3. For the considered position, K, of the crank, the 
movement of the valve was found to equal the distance Q k, 
and hence the opening of the steam port S (Fig. 1) for ad- 
mission of steam is equal to ka, and that of the port S' for 
exhaust equal to ka' ; these distances being equal to the move- 
ment Qk, minus the respective laps. The port openings are 
therefore represented by the shortest distances between the 
centre line of the crank and the lap circles L or l } respectively. 

Had the point Q\ instead of Q, been selected, the result 
would have been the same, since Q k equals Q'k'. 

To prevent any misunderstanding it may be here stated 
that Fig. 3 now represents two distinct diagrams, one show- 
ing the position of the eccentric, I, in relation to the crank, 
OK, when on its centre, and another in which the point Q 
figures and which will be called the "valve diagram." The 
latter can be obtained from the former by making the angle 
XO Q = angle YOT, or also, by rotating the first diagram- 
through a right angle, when the eccentric will arrive at J," 
and subsequently inverting it. This last mode of converting 
one diagram into the other, can very conveniently be per- 
formed by means of a piece of tracing paper, viz., by tracing 
the diagram of crank and eccentric, KOI, turning this trac- 
ing over and setting the crank-line, OK, in a vertical position, 
pointing downwards. The position of the eccentric, I, will 
then locate the point, Q, whose mate, Q\ is diametrically 
opposite. 

In order to know the direction of the valve-movement we 



12 



must consider that the valve will be in its neutral position 
when the crank passes through Q or Q\ the distance of Q 
from the crank-line being zero. As the crank moves from Q y 
through K' to Q, the movement of the valve is positive, and 
while moving from Q to Q' this movement is negative. 

When the engine is required to rotate in an opposite direc- 
tion, the position of the eccentric will be 1°, in relation to 
the crank, OK ; and the valve diagram will be reversed or 
left-handed. 

The valve diagram as explained affords an easy means of 
studying the influence of form and motion of the valve on 
the distribution of steam. The most important positions of 
the crank, as regards the admission and the exhaust of steam, 
•are shown in Fig. 4. When the crank is on its centre, at K, 




Fig. 4. 



13 
the distance ka shows the opening of the port for admission,, 
or the lead on the steam side, and ka' represents the opening- 
of the port for the exhaust, or the lead on the exhaust side. 
The valve attains its maximum movement when the crank is. 
at N s forming right angles with the line OQ; hence the 
greatest port-opening for admission of steam is equal to Ob. 
This distance may be greater than the width of the steam 
port, showing that the outside edge of the valve overruns- 
the inside edge of the port, allowing an unobstructed admis- 
sion of steam. The word " port-opening " will be used 
independently of the width of the steam ports, and may 
therefore exceed the latter. As the crank proceeds, the port- 
opening will be reduced again ; and when the crank arrives at 
E, the port, S, (Fig. 1) will be closed and the admission of 
steam cut off; the valve movement being equal to the outside 
lap. At C the valve movement will be equal to the inside 
lap, hence the port, S', (Fig. 1) will be closed, and com- 
pression will commence. OQ is the position of the crank 
when the valve passes its neutral position At B the negative- 
movement of the valve equals the inside lap and the port, 3, 
is opened for exhaust. At A the port, S' t will be opened for 
admission of steam for the following stroke. For the crank- 
angles X\E' ', C\B\A\ the results will be merely a repetition 
of what has already been said, with this difference only : that 
the ports exchange their respective functions. The lead angle 
for admission is shown in the diagram by the angle A' OK ot 
AOX, and that for exhaust by the angle JR' OK or BOX. 

This diagram may perhaps seem, to a beginner, somewhat 
complicated, especially as it is necessary for some crank 
angles to extend the crank-line over the centre, ; and fur- 
ther, as it requires some practice to determine which of the- 
two ports is being opened or closed. It can, however, be so 
drawn that the operation of the valve on each port may bo 
represented separately, as in Fig. 5, relating to the port, 3* 



14 




(Fig. 1) and in Fig. 6, relating to the port, S 9 . By a com- 
parison with Fig. 4, these diagrams may be readily under 
stood without further comment. These diagrams indicate 
the port to which they relate to be opened, for either admis- 
sion or exhaust, as long as the crank-line is outside of the 
respective cross-lined surface and each of them may again 
be regarded as combining two diagrams ; one for admission, 
and the other for exhaust. 

These two diagrams, Figs. 5 and 6, can readily be combined 
by drawing the upper half of Fig. 5 and the lower half of 
Fig. 6 in full lines and the rest in dotted lines. Some of the 
following diagrams will be executed in that manner ; and it 
is only to be remembered that the crank-angles for cutting-off 
and for release (relating to the port on the steam side of the 



15 



<rP 




piston) will always be tangent to solid circles, while the 
crank-angles for compression and the beginning of admission 
(relating to the port on the exhaust side of the piston) will be 
tangent to the dotted lap circles. 

When the valve possesses no inside lap, the inside lap-circle 
shrinks into the point, Q, and hence Qand Q y are the positions 
of the crank where the exhaust is closed for one port and at 
the same time opened for the other port. Sometimes the valve 
may be constructed as shown, in Fig. 7, possessing the con- 
verse of an inside lap. The distance, cd, is properly called 
"negative lap." Some engineers call it "inside lead," 
others, again, "inside clearance/ ' but both these terms are 
apt to lead to confusion, for "lap" is measured when the 
valve is in its neutral position; "lead" is measured when 



16 




•tf'JV 



Fig. 7. 

the crank is on its centre ; and " clearance " is the conven- 
tional name for the space left between the steam-valve and 
the piston at the end of its stroke. To find the opening of the 
port for exhaust in such a case, the distance, cd, must evi- 
dently be added to the movement of the valve, and hence the 
opening is not Jca 7 (Fig. 3) but Jca" and- the diagram repre- 
sented in Fig. 5 becomes like Fig. 8, which shows that in 




17 

this case the exhaust remains opened for more than a semi- 
revolution of the crank shaft. 

It has been stated before that the greatest port-opening of 
the valve sometimes exceeds the width of the port. The dia 
gram shows without difficulty the angles for which the port 
-remains opened to its full width. We sweep a circle, s, from 



/ 



JJL -Sk--..H»^fc: t „ 




Fig. 9. 

"^ (Fig. 9), the distance of which from the lap circle, 
Z, is equal to the width, w, of the port. For the two 
crank angles, OK and OIC, which form tangents to this 
circle, the port -opening will be equal to the width of the port, 
and between K and IP it will be in excess of the latter. 

Or, it may happen that the greatest port-opening for ad- 
mission is less than the width of the port, which then will 



18 

become fully opened for the exhaust only. The greatest 
port-opening for admission should, however, never be less 
than 75 per cent, of the width of the port. 

The manner of comparing the motion of the crank with 
that of the piston will be explained later. 

The Angle of Advance. 
On horizontal engines the centre-line 01 of the eccentric 
forms, as already mentioned, an obtuse angle with the centre 
line OiT(Fig. 2) of the crank. This relation will be modified 
when both the connecting rod and the eccentric rod are not 
operated in parallel directions, as may be the case with beam 
engines, or where a rocker or some other lever connection is 
interposed, for instance as shown in Fig. 10. But no matter 




<4'r 

Fig. 10. 

whatever the general arrangement of the engine may be, the 
centre line of the eccentric forms most always approximately 
right angles with the eccentric rod when the crank is in line 
with the connecting rod, or "on its centre.' ' It should, how- 
ever, never be precisely at right angles, but always more or 
less in advance of this position, in the direction of the rotation 
of the crank shaft, for else the valve would be in its neutral 
position at the beginning of the stroke ; and since the port 



19 

on the steam side is required to be opened, the eccentric 
must be advanced to make the lap and lead. The angle 
measuring this advance, namely, the angle formed between 
the eccentric line 01 (Fig. 10) and the line OY drawn at 
right angles to the mean direction of the eccentric-rod, 
is called the ''angular advance" or "angle of advance." 
The mean or neutral direction of the eccentric rod may be 
determined, in the drawing, by transferring the end 7 of the 
same into the centre of the crank-shaft, and is shown, in 
Fig. 10, by the dotted line 0P°. Whenever the end P of 
the eccentric-rod is guided in an arc, as in Fig. 10, this con- 
struction does not give a perfectly accurate result, but it will 
be sufficiently accurate for practical purposes, as the valve 
performs its principal operations when near its central posi- 
tion. 

The "angular advance" plays an important part in the 
action of the slide valve and shall be denoted by the letter 6, 
as has already been done in the former diagrams. 

In order to investigate the operation of its valve motion, 
any type of engines may be compared with a horizontal 
engine having the same slide valve, with equal travel, and 
equal angular advance. The valve diagram can therefore be 
drawn by first sweeping a circle for the path of the eccentric 
and then carrying in it the angle of advance from the hori- 
zontal centre-line in a direction opposite that of the rotation 
of the crank. From the points thus obtained in the circum- 
ference of the circle the lap circles have to be swept to 
complete the diagram. 

It need hardly be mentioned that in cases where the motion 
of the eccentric-rod is transmitted to the val^e by rockers or 
levers of unequal leverage the radius of the "path of the 
eccentric" of the diagram should not be made equal to the 
throw cf the eccentric, but equal to one-half the travel of the 
valve. 



20 

In designing engines other than such where the connect- 
ing-rod and eccentric-rod are operated in a parallel direction, 
beginners often find difficulty in drawing the eccentric in its 
proper position to the crank, the angle of advance being 
given. The course to be pursued in such cases is as follows. 
Draw the crank OK (see Fig, 10) on one of its dead centres, 
and then draw the line F, at right angles to the mean direc- 
tion, OP°, of the eccentric-rod and cany the angle of ad- 
vance, S, from the last mentioned line, in the direction of the- 
rotation of the crank-shaft. The resulting line determines 
the angle of the eccentric, but it must be decided yet whether 
I or I 9 (see Fig. 10) is the correct position. This can be 
done by assuming the movement of the crank to proceed, and 
considering which of the two points J or J' would cause the- 
valve to open that port that will admit the steam on the pro- 
per side of the piston. 



Width of Ports. 

The sectional area of the steam-ports should be made so 
large as to impede the ingress and egress of the steam as little 
as possible, and yet, on the other hand, so small as not to 
unduly increase the amount of clearance. By a series of care- 
fully conducted experiments, it was found that the greatest 
velocity of the steam in the passages should be from 8,000 to 
9,000 feet per minute, in small engines and such of a low 
pressure and low mean piston speed, and from 9,000 to 10,- 
000 feet in large ones and such of high piston speed. 

To compute, from these data, the area of the ports in any 
one special case, the maximum velocity of the piston must be 
found, which is equal to that of the crank pin, and hence 
equal to the product of the number of revolutions per minute 
into the periphery of the crank-pin circle. The conventional 
measure for the speed of engines, however, is the mean 



21 

^velocity of the piston, which is equal to the number of revolu- 
tions into twice the diameter of the crank-pin circle. The 
proportion of the mean speed to the maximum speed is there- 
fore as two diameters to the circumference of a circle, or ap 
proximately as 7 is to 11 ; and hence the maximum speed of 
& piston is equal to its mean speed multiplied by the fraction 
y. The ratio of this speed to the allowed maximum speed of 
the steam in the passages furnishes the ratio of the area of the 
steam-ports to the area of the piston. 

After the area of the ports is thus determined and a suitable 
length of the ports is selected, their width is found by divid- 
ing the area by the leng:h. 

It is well to make the length of the ports as large as prac- 
ticable to procure a small width ; for the travel of the valve is 
•directly dependent on this w T idth, and reducing the travel 
Implies a reduction of the amount of work to be expended in 
overcoming the friction of the valve. 

Some attention is to be paid to the dimension of the exhaust 
port. If it is made wider than necessary the valve must be 
made longer than otherwise, and a needless increase of press- 
ure and a consequent loss of work from friction will result. 
If, on the other hand, it is made too narrow, the valve will, at 
the end of its stroke, cover it more than desirable and choke 
the exhaust. 

In order to find the proper w T idth of the exhaust port we draw 
the steam-port S, (Fig. 11) and over it one leg of the valve with 
the proper inside and outside lap, first in its neutral and then in 



h 




22 

its extremest position, removed towards the exhaust port. The 
distance ab is the outside lap, be the width of the port, cd the 
inside lap, and ae or dgis equal to half the travel of the valve.. 
The width of the bridge, cf, should be as narrow as possible, 
i. e. just wide enough to insure a perfect casting and sufficient 
strength to withstand the steam pressure. But if then the 
distance ef should happen to be less than a quarter inch, the 
bridge must be made wider accordingly, to prevent the possi- 
bility of a leakage of steam from the steam chest to the exhaust 
passage. If the bridge, on the other hand, is wider than neces- 
sary to prevent leakage, the inside corner may be rounded, as 
indicated in Fig. 11, to facilitate the passage of the exhaust 
steam. The other edge, h, of the exhaust port should have a 
distance from the inside edge of the valve, g, equal to the 
the width of the steam port be, or not much less in order that 
the exhaust steam need not contract beyond the area of the 
steam passage when entering the exhaust port. After the 
exhaust passage is thus located, the second port can be drawn 
on the other side and the dimensions of the slide valve can be 
determined by drawing the other leg of the same over the 
second port — as shown by dotted lines in Fig. 11. 



Velocity of the Valve. 

A few words on the velocity of the valve in the different 
stages of its tmvel and on the rapidity of the opening and 
closure of the ports will not be out of place. 

The valve moves with increasing velocity from the begin- 
ning of its stroke until it reaches its neutral position, where 
its speed is at the highest rate. Thereafter the speed again 
diminishes until at the end of its stroke the valve comes, for 
an instant, to a stop, and then reverses its motion. If we 
assume a perfectly uniform rotation of the crank-shaft, the 
velocity of the valve will always be in direct proportion to 
the leverage under which the eccentric is acting upon the 



23 

valve ; and as the selection of a scale for a graphical repre- 
sentation of this velocity is quite optional, we may choose th e 
length of this leverage directly for the representation of this 
velocity. The maximum velocity will then be expressed by 
the length, 01, (Fig. 3) and this occurs when the eccentric is 
at right angles, or, in other words, when it passes the line, 
Y When, however, the eccentric is, for instance, at T the 
actual leverage is equal to 0y\ and the corresponding velo- 
city is expressed by this length. In the foregoing it has been 
demonstrated that the triangles, OQk' and OFy' (Fig. 3) are 
equal ; and hence Ok = 0y\ From this it follows that the 
diagram can also serve to answer the question of the com 
parative velocity of the valve for any stage of the stroke. 
All that is needed for this purpose is a line drawn from the 
point, Q, of the diagram, at right angles to the respective* 
crank-line, 0K\ when the distance of the foot, k, of this line, 
from the centre, 0, is the equivalent of the velocity of the 
valve while the crank passes the the line, 0K\ To find the 
velocity of the valve at the point of the cutting off, when the 
crank is at OE, (Fig. 12) we draw the line, Qk at right angles t a 
OE. The distance, Ok, then represents the sought velocity, 




Fig. 12. 



24 

rand since the point, k, is the point of contact of the line, OB, 
and the lap circle, L, we may remember, as a rule, that the 
distance of this point of contact from the centre, 0, always 
represents the velocity of the valve at the moment of the cut- 
ting-off. 

The rapidity of the closure of the port depends not on 
this velocity alone, but also on the velocity of the valve from 
the time it begins until it completes covering the port. Fig. 
12 shows that, previous to the cutting off, when the crank is, 
say, at K\ the velocity of the valve, Ok' is smaller than Ok. 
This result is consistent with the preceding consideration, 
according to which the motion of the valve is increasing be- 
fore reaching the centre of its travel, and as the valve cuts 
the steam off before that point by virtue of the lap, the ve- 
locity is evidently less before the time of the cutting off than 
it is at that moment. It will also be seen now that the sharp 
ness of the cutting off suffers as the lap is enlarged, and for 
this, as well as for other reasons, the throw of the valve should 
be increased as more lap is added. 

The length of the steam ports favors the sharpness of the 
cutting off, and therefore, if like results are desired, the ve- 
locity, in other words, the throw, can be lessened in the same 
ratio as the length is increased. It should, however, at the 
same time be remembered, that an increase of the length of 
the ports implies a proportionate decrease of their width, if 
the same area is to be preserved. 



25 



Practical Use of the Diagram. 

The diagram explained in the foregoing can be applied in 
practice to any question relating to the construction of slide 
valves. The question simply resolves into the problem of 
drawing the diagram in accordance with the given data, and 
the sought dimensions or results can be taken from it. Be- 
ginners are advised to sketch a diagram of the general form 
and to mark in it those dimensions that are known or given. 
The execution of a drawing to scale will then oner no diffi- 
culty. 

The diagram being drawn, it is easy to proportion the valve 
gear accordingly. The diameters of the " eccentric path Z" 
of the diagram represents the travel of the valve, and the 
throw of the eccentric has to be made to correspond. The 
proper dimensions of the valve can be found as shown in a 
previous chapter (width of ports) from the laps given by the 
diagram. If a valve thus proportioned is properly set, its 
operation will correspond with the indications of the diagram. 

In setting a valve two things are to be taken into considera- 
tion : first, the length of the eccentric rod or of the valve 
stem, and second, the angular position of the eccentric on the 
crank-shaft. To accomplish the first point it is necessary 
that either the length of the eccentric rod or else that of the 
valve stem is made adjustable. It is an object to adjust this 
length with as few turns of crank shaft as possible, especially 
on large engines. To this end turn the eccentric until the 
valve stem is in one its extremest end-positions, say as far 
towards the crank-shaft as it will go. Then adjust the valve 
according to the greatest port-opening shown in the diagram, 
or, in other words set it so, that the distance of its outside 
edge from the outside edge of the respective steam-port (S in 
Pig. 1) is equal to the greatest port-opening of the diagram. 



26 

This "being done, set the crank on its dead centre with the 
piston nearest the crank-shaft, and, holding it firmly in this 
position, advance the eccentric until the proper port (S in 
Fig. 1) is opened by the valve to an extent equal to the lead 
required, and tighten it on the crank-shaft. If now the crank 
is turned on its other dead centre, the lead for that side will 
be found to slightly differ from the lead required, owing to 
the influence of the angularity of the eccentric rod, which, it 
will be remembered, we had omitted to consider until now. 
To correct this difference, the eccentric rod (or the valve rod, 
as the case may be) must be lengthened by one-half the dif- 
ference observed ; the remaining half of this difference is 
subsequently removed by changing the position of the eccen- 
tric on the crank-shaft, care being taken that the crank re- 
mains on the dead centre to which it was adjusted last. 

In this way the setting of the valve requires the crank to 
be set but once to each of its dead centres. 

A later chapter will show that sometimes the lead of the 
fore-stroke is required to differ from that of the return- stroke. 
The given rule will be equally applicable for setting the valve 
in such a case. 

In the following we shall leave out of consideration the in- 
fluence of the obliquity of the connecting rod ; in other words, 
we shall consider the connecting rod to be infinitely long. 
The position of the piston corresponding to any one crank 
angle may therefore be found by drawing the crank-path to a 
reduced scale, and projecting the crank pin centre on its hori- 
zontal diameter or any other horizontal line. The crank- 
circle may sometimes be drawn to a scale to make it coincide 
with the path of the eccentric, as will be shown in some of 
the following problems. 

Problem L A small steam-engine of 7" stroke is provided 
with a valve of §" outside lap and T V inside lap. The travel 
of the valve is 1£" and the lead is adjusted to^". What 




27 

-will be the angular advance, how much expansion and cush- 
ioning will take place, when will the exhaust commence, and 
how much will the outer edge of the valve overrun the inner 
edge of the port if the width of the port is T y. 

At first we sketch a diagram of the usual form (Fig. 13) 



ii*! 3" 




Fig. 13o 

and mark in it those dimensions which are known, and in 
referring to this sketch the course of construction will plainly 
be seen. The circle Z, representing the path of the eccentric, 
is given by its diameter, equal to the travel of the valve, and 
•can at once be drawn. The lap circle L should be T y above 
the line OX, and its radius is known to be §", hence its centre 
Q must be T V above OX. A horizontal line £$" above OX 
will therefore intersect the circle Z in Q> and both lap circles 
may now be drawn. The crank-positions E, C, B and A, 
will respectively indicate the commencement of expansion, 
compression, release and admission. If we finally draw the 
crank circle Z' at a reduced scale and project the crank- 
centres on the line HIT, which represents the stroke of the 
piston, we find the cutting-off to take place at e, which is 4£ ,f 



28 

of the stroke. Cushioning commences at c and the release at 
r, respectively |" and \ n before the termination of the stroke. 
The largest port-opening can be directly measured in the 
diagram and will be found to be | "which is ^ n above the width 
of the port. The angular advance is given by the angle 6. The 
point of admission is here and in most of the following pro- 
blems practically at the end of the line HH. 

Problem II It is required to find the outside and inside 
lap of the valve for an engine of 20" stroke, to cut the steam 
off at | stroke (15") and to cushion the exhaust steam for 2". 
The width of the ports is §", the travel of the valve 4", and 
no lead is to be given. 

The points for cut-off and cushioning being given, the cor- 
responding crank angles can readily be found, and they may 
be regarded as given, in the sketch Fig. 14. The lead, marked 
*by a dart, should be made = 0, in other words, the lap circle 
should rest on the line OA. If we draw the crank circle to ^ 
scale it will coincide with the path of the eccentric. 

It may be remarked here that Fig. 14 and most of the fol- 
lowing valve diagrams are not made to full size, as they would 




Fig. 14. 






29 

take up too much space. In practice, however, the eccentric 
path should always be drawn to full size to ensure a sufficient 
degree of accuracy. 

The construction will be as follows. Sweep the circle Z with 
2" radius. Mark on the line HH, representing the stroke of the 
piston, the points e and c> for cutting off and for cushioning, and 
obtain, through vertical projection-lines the corresponding 
crank-angles OE and OC. Both the lines OE and OA must be 
tangent to the lap circle, the centre of which will therefore be 
located on the circle Z by a line bisecting the angle EOA. 
The lap circles L and I can now be swept, from Q, tangent, 
resoectively, to OE and C, and the crank angle for release, 
OB, can subsequently be drawn. The radii of the lap circles 
are the required laps, which are found 1" and \ n respective- 
ly. The angular advance is 6= QOA, the release, at r, takes 
place less then 1" before the end of the stroke, and the largest 
port-opening is = 1 /7 , which is ^" above the width of the 
port, 

Problem III. An engine of 16" stroke is required to cut 
off at 10" and have a cushioning of 1|". The largest port 
opening is required to amount to f" and the lead to £". 

The crank-angles for cut off and for cushioning, the lead 
and the greatest port-opening are known, as indicated in the 
sketch Fig. 15, and the diagram can be drawn as follows : 

Sweep the crankpin-circle Z' to reduced scale, and under 
it the line IIH for the stroke of the piston. Mark on it the 
points e and c for expansion and compression and find the 
corresponding crank -angles OE and OC. The lap circle must 
be tangent to OE; it must be £" above the line OX and must 
be I" from the centre 0. Therefore we draw a horizontal 
line h £" above OX and a circle s, from 0, with a radius of 
J". The required lap circle must touch the lines OE, h and 
the circle &, and its centre can be found by bisecting the angla 



30 



EV^m? 



/ 
/ 

r 



.JSLy.-g- — A— 



\ 




Fig. 15. 

formed by OE and h and finding hy trial on this line a point 
Q from which the required lap circle can be drawn. The 
circle Z swept from through Q forms the path of the eccen- 
tric. A circle from Q tangent to G gives the inside lap 
circle, which in this case is found to represent negative lap, 
since the crank-angle for compression should be tangent to 
the dotted half of the circle. The necessary dimensions are 
found to be : travel of the valve 4 J", outside lap l^ n , inside 



lap, negative, £ /; . 
end of the stroke. 



The release takes place 2£" before th& 



Problem IV. The stroke of an engine is 21 " the travel of 
the valve 3J", the outside lap is §" and the inside lap= 0". 
The width of the ports is | /; and the lead £". By this valve 
the cutting-off is effected at about 16 J" of the stroke. It is- 
required to change the cut off to 14" of the stroke by chang- 
ing the throw of the eccentric, leaving the lead angle the- 
same as it was before. 

We first draw the diagram for the original travel of 3£"' 



31 

(Fig. 16) to obtain the lead angle A OX. Next we construct 
the crank-angle OE for the cutting off at 14". Knowing now 
that the lap circle of the new diagram, the radius of which ia 



£ ;..j* 




Fig. 16. 

to be j", must touch both the lines OA and OE, we draw 
two lines at a distance of §" from, and parallel to, these two 
lines, and the point of their intersection, Q', is the new cen- 
tre of the lap circle. Its distance from Ois the required throw 
of the eccentric, which is If". The greatest port opening for 
admission will be | /y which is £" short of the width of the 
port. The angle of advance, the degree of cushioning and 
the point of release can subsequently be found from the dia- 
gram in the usual way. 



32 



The Scope of the Common Slide Valve. 

The most primitive form of the slide valve is that without 
any lap, lead or angular advance, by the operation of which 
the piston will be exposed to the full steam-pressure on one 
side, and to exhaust on the other, for the entire duration of 
the stroke. But the instant the crank passes the centre, the 
conditions are suddenly reversed and that end of the cylinder 
which was just being filled with steam is now brought in 
communication with the exhaust passage. It is evident that 
all the steam filling the cylinder cannot instantly escape 
through the opening which at first is very small, and a con- 
siderable back-pressure during the first part of the stroke will 
"result. Besides, the expansive power of the steam is not 
utilized at all, but is prodigally wasted. To obviate tlicsj 
losses lap must be given to the valve and the eccentric must 
be advanced. But the common slide valve should, under 
ordinary conditions, never be used to close the ports before 
half stroke, for a reason that will at once be seen if we ex- 
amine the operation of a valve with earlier cut-off. Suppose 
it was required to construct a valve with 4" travel, to cut off 
at J of the stroke, with no lead. We draw the circle Z (Fig. 
17), determine the crank-angle OE for £ of the stroke, and 
bisect the arc EA to obtain the centre Q of the lap-circles # 
The outside-lap circle can now be drawn at once. Sup- 
posing the valve to be without inside lap, the cushioning 
on the one and the release on the other side of the pis- 
ton will begin when the crank passes the line OQ. Re^ 
garding the line Hffl as representing the stroke of the 
piston we find that compression and release begin at x ; the 
expansion, from e to x y is curtailed by an early exhaust, and 
the compression from x to H is excessive. If we endeavor 
to delay the exhaust by giving inside lap, in order to prolong 



33 




Fig. 17. 



the expansive action of the steam, we find, by drawing the 
lap circle I and the crank-angles C and OR, that the amount 
of cushioning will be increased. On the other hand, if we try- 
to diminish the degree of compression it will be necessary to 
apply negative inside lap, which may again be represented by 
the circle I. The cushioning will then commence at OR ; 
but it will be noticed that this measure will hasten the release 
which then takes place at C. We see from this that with 
no inside lap, both the release and the cushioning begin much 
sooner than desirable, and that any change for the sake of 
delaying the one causes a hastening of the other. 

Another very disadvantageous point will be noticed. The 
greatest port-opening indicated by this diagram is but little 
more than ^ of the travel of the valve. Hence it is clear that 
a plain slide valve so proportioned as to cut off at ± of the 
stroke produces but a small opening for the admission of the 
steam. 



34 



Irregularities of the Crank Motion. 

The relation between the movement of the piston and that 
of the crank- shaft is in fact somewhat different from that con- 
sidered in the preceding pages. When the crank is at right 
angles (see Fig. 18) the piston is not in the middle of its 







Fig. 18. 

stroke but is nearer to the crank -shaft, on account of the 
obliquity of the connecting-rod. At any other crank-angle 
the obliquity of the connecting-rod will likewise affect the 
position of the piston, though to a less extent, the nearer the 
crank approaches either of its dead centres. 

A circular arc swept from the crosshead end of the connect- 
ing-rod through the crank-pin centre will intersect the hori- 
zontal diameter of the crank circle in a point having the same 
relation to this diameter as the position of the piston has to its 
stroke. If it is required to find the true position of the piston 
for a given crank-angle, instead of projecting the crank-pin 
vertically, we should project it upon the horizontal diameter 
by a circular arc ; and when the position of the piston is 
known the corresponding crank-angle can, be found in the 
same way. The radius of this projection arc should be equal 
to the length of the connecting-rod, measured on the same 



35 

scale to which the crank-pin circle is drawn, and its centre 
should be located on the line of movement of the crosshead. 

These variations from the previously considered ideal move- 
ment of the piston will evidently gain in extent as the con- 
necting-rod is shortened, and the latter should, for this as for 
other reasons, rarely be made shorter than 4£ times the length 
ofihe crank. On horizontal engines it is usually equal to 5 
times the crank. 

It will now be understood that the piston runs ahead of its 
ideal movement on its " forward stroke ' ' (that is when the con- 
necting-rod is strained on compression, ) and stays behind the 
same during its return stroke. The result will be the admis- 
sion of more steam on the fore-stroke than on the return- 
stroke, and hence an unequal development of work at both 
strokes. This difference will be augmented by the presence 
of the piston-rod, which diminishes the active surface of one 
side of the piston, except where an extension of the piston- 
rod passes through a stuffing box on the back head of the 
cylinder. 

The movement of the valve, in its relation to that of the 
crank shaft, is subject to similar inequalities, owing to the 
oscillations of the eccentric-rod, and the valve will be drawn 
more or less towards the crank-shaft. However, as the eccen 
trie-rod is generally very long in relation to the throw of the 
eccentric, these variations are small, and besides their effect 
can easily be neutralized by slightly lengthening the valve 
stem, which measure will cause the greatest port opening at 
the fore-stroke to be a trifle less than that of the return-stroke. 
This correction is achieved by the practice of setting the 
valve by the lead, or, in other words, by adjusting the valve 
stem while the eccentric-rod is in an inclined position. 

The inequality of the cut-off, occasioned by the obliquity 
of the connjcling-rocl, can be rectified by giving the valve a 
different lap for each of the two ports, but this rectification is 



36 

attained at the expense of an equal lead, and is therefore 
seldom taken advantage of in practice. 

An equalization of the compression can be effected by fol- 
lowing the same course respecting the inside-lap. 

On engines that are provided with valves having equal lap 
on both sides, a rectification of the point of cut-off can be 
accomplished by lengthening the valve-stem. This operation 
is equivalent to an enlargement of the laps relating to a posi- 
tive movement of the valve and a diminution of those relat- 
ing to a negative movement ; it will therefore destroy the 
equality of lead, and it will more than rectify release and 
cushioning. In this case the neutral position of the valve is 
no more a central one, but is as shown in Fig. 19 where the 
laps ab, and c'd' exceed the laps a'b' and cd, the former per- 
taining to a positive, the latter to a negative movement. 




Problem F. It is required to find the exact positions of the 
piston for the cutting off, the release and the cushioning of 
the engine considered in Problem L, if the connecting-rod is 
16" long. 

The diagram can be constructed as shown before, but in- 
stead of drawing vertical projection-lines from the crank-pin 
centres E, C, and R, circular projection-lines are to be 
used, of a radius equal to 16", measured, however, on the 
scale to which the crank-pin circle has been drawn. The 
resulting points can now easily be transferred upon two 



37 

separate lines, for the fore and the return stroke, as shown in 
Fig. 20. 

JFore Stroke 




H^c' 



Fig. 20. 



A comparison between the fore and the return stroke will 
be easier if the lower half of the diagram is drawn in com 
bination with the upper half (see Fig. 21). This form of the 
diagram shows very plainly the amount of the variation, as 
compared with the results of our first method of projecting by 
vertical lines. 

Problem VI. — The case of Problem II. is to be solved for 
equalized cut off and equalized compression, if the connect 
ing rod is 45" long and the lead for the fore stroke £%''. The 
solution of this problem involves the drawing of two distinct 
diagrams, one resembling Fig. 5, the other Fig. 6, both dif 



38 



fering by their lap-circles, since the lap for both ports will 
differ. Both diagrams can, however, readily be drawn in 
one figure 



^z> — 




1 Fore Stroke \ j, 




h ^ > i 

; R eturn Stroke \ 

h - -■*—■ 



-4-^6-0 



e \c\rH 
<U-4 



> d c't'H 



Fig. 21. 

We draw in Fig. 22 the crank circle Z, as we did in Fig 
14, and mark on the horizontal diameter the points e and e f 
for the cutting off, and c and c' for the compression. The 
crank-angles E, E' } C, and C are then found by sweeping 
the projection arcs of the proper radius, and it will be seen 
that the corresponding crank-angles for both strokes are no 
longer diametrically opposite. Since the lead for the fore 
stroke is given, the diagram for that stroke can be drawn 
by drawing the horizontal line 7i, -^" above OX, and bisect- 
ing the angle formed between this line and the line OE, to 
obtain the point Q on the circle Z, which, it will be remem- 
bered, serves in this case for both crank path and eccentric 
path. By drawing a line from Q through we find the 
point Q r , which happens to coincide practically with the 
point C in this case. Now we can draw the lap-circle L, in 
full line, from Q to touch OE, and its mate, in dotted line 
from Qi ; also the lap-circle L f , from Q' to touch 0E' y and 



39 




e cr \ X 



y*% 



Fig. 22. 

its mate from Q. Next we draw, in clotted line (relating to- 
compression, on the exhaust side of the piston), the lap- circle 
V from Q, and its mate from Q' in full line. The last lap- 
circle, from Q' tangent to OC, remains yet to be drawn, but 
its radius is zero. The diagram now shows that the lead for 
the return stroke is more than | 7 ', which considerably ex- 
ceeds that of the fore stroke, and due attention must be paid 
to this circumstance in adjusting the valve on the engine. 
In fact, herein lies the greatest practical difficulty in rectify- 
ing the cut-off as shown, since most engineers are accus- 
tomed to the practice of setting the valve to equal lead, and 
this valve, when so adjusted, will distribute the steam to less 
advantage than if no rectification would have been attempted. 
In other respects there can be little objection to this meas- 
ure ; for, notwithstanding the considerable lead on one side. 



40 

the space which the piston is compelled to travel against the- 
steam after admission is but a very small fraction of the 
stroke. The crank angles and positions of the piston for the 
release may be found without difficulty in the diagram. 

Those circles swept from Q in full lines represent the laps 
of that side of the valve that covers the port, S, (Fig. 1> 
while those swept in full lines from Q' give the laps for the 
other side of the valve. The inside lap of the steam passage, 
S, is zero. 

Problem VII. The valve considered in problem V is to be 
rectified for the cut-off by lengthening the valve stem. How 
much has the valve stem to be lengthened, and what effect 
will this measure have upon the lead and the exhaust ? 

The eccentric-path, Z, the crank-path, Z\ and the point, 
Q, can be copied from Fig. 20. The proposed lengthening of 
the valve stem will hasten the cutting off at the fore stroke 
and delay that of the return stroke, and the cutting off will 
take place at the intermediate piston position, e°, (Fig. 21) 
obtained by a vertical projection.* After copying this point 
from Fig. 21, we can find the exact crank angles, E and 
E' y by the projection arcs of the proper radius. The out- 
side lap circles can then be swept, and their distance from the 
original lap circle equals the length by which the valve stem 
has to be lengthened, and which is found = about T V'- The 
lead for the fore stroke will be zero, and that for the return 
stroke = | ". 

In lengthening the valve stem by T y the inside lap for the 
fore-stroke becomes ^"-\- xV — i" an( * tnat f° r tne return- 
stroke — i jV 7 — T J " = 0. We can accordingly finish our 
diagram and determine the crank angles and piston -positions 

*This last assertion is theoretically incorrect, for the point of cut- 
off, af'er the proper rectification will not precisely coincide with the 
..joint, e° y but the difference is practically inappreciable. 



41 



for cushioning and release, and a comparison of the resulting 
points c, c' t r and r' with the corresponding points of Fig. 20 
will show that the measure resulted in an over-correction of 
release and cushioning. 

v— E 

V z 

III *L e 

. M * 




PART II 



LINK MOTIONS. 



45 



Link Motions. 

It is often required that steam engines should be con- 
structed to run either forward or backward, as is the case 
with locomotives, marine engines, hoisting engines and others. 
Valve gears, by which this end can be accomplished, are called 
"reversing gears," and most of those now in practical use 
admit of a variation of the degree of expansion. The almost ex- 
clusive use of "Link Motions " for this purpose makes it of the 
utmost importance that steam- engineers should be perfectly 
familiar with the peculiarities of these most ingenious devices, 
especially since they can be proportioned to correct the dif- 
ference between fore and return stroke occasioned by the 
obliquity of the connecting rod. 

The number of different link motions and other reversing 
gears admitting variable expansion is not small, but it seems * 
that the two first inventions (the Stephenson, and the Goocli 
link motion) have stood the test for fitness and durability 
better than any of their followers. We shall therefore con- 
sider the theory of these two gears only, but it may be re 
marked that the operation of nearly every other reversing- 
gear can, in some measure, be compared with that of either 
of these two link motions. 



46 



The Stephenson Link Motion. 

This link-motion, the general arrangement of which is 
shown in Fig 24, is the one most generally used, especially 
on locomotives. 




Fig. 24. 

Two eccentrics I' and I" are secured to the crank-shaft, 
and the ends P' and P ,f of the rods of these eccentrics are 
jointed to the " expansion link" or commonly called, briefly, 
"link," which is a curved frame of wrought iron or steel, 
containing a sliding block B that is fitted into the slot of the 
link and is attached, by a joint, either directly to the valve- 
stem V or else to a rocker by which the slide-valve is moved. 



47 



Two of the more usual forms of the link are shown in Figs. 25 
and 26. The slide-valve is a single one of substantially the 
same shape as is used in stationary engines. The link is 
suspended at the stud D (Fig. 24) by the suspension -rod, or 
"hanger," GD f and can be lowered or raised by means of 
the "suspension lever" FG, which is connected, as the 
drawing shows, to the reversing-lever U, the latter moving 
on a notched arc on which it can be fixed at anj^ desired 
position. The hanger GD may be attached to the lower end 
of the link (see D in Fig. 25 or P" in Fig. 26) or, to a stud 
secured to the centre of the link by a bridge or bracket (Fig. 
24.) Sometimes the suspension lever FG is located below 
the link, placing the hanger in an inverted position. 





Fig. 25, Fig. 26. 

The eccentric I' is called the forward eccentric, and is 
fastened to the crank-shaft in such a position that its operation 
upon the link-block B would produce a forward-rotation of 
the engine. The other eccentric, J", is the backward or 
hacking eccentric, and would cause an opposite rotation of 
the engine if conected with the link-block. 



48 

When the crank-shaft rotates, these two eccentrics will 
transmit to the link a movement of a very peculiar character 
being both oscillating and reciprocating at the same time. 
By lowering or raising the link either of the two eccentrics 
can be brought in action, and thus the engine may be run 
forward or backward at pleasure. The link can, however, 
also be used in intermediate positions or " grades" when 
both eccentrics more or less influence the movements of the 
valve, and it will be the chief object of our future investiga- 
tion to find the law of the movement of the valve for any 
grade of the gear. 

The eccentric-rods may be used either "open" or^ 
"crossed." They are called "open" when they are con- 
nected with the link as represented in Fig. 27, and "crossed" 
when they are arranged as Fig. 28 shows. To make this 
distinction both eccentrics must be inclined towards the Unk y 





Fig. 27. 



lig. 28. 



for when the crank-shaft is turned half a revolution, to bring^ 
the eccentrics on the other side, the link-motion with open 
rods will appear crossed, and vice versa, and care must be 
taken, therefore, not to confound both types. 

When the link-block is not attached directly to the valve- 
stem, but to a rocker, by which the direction of motion is 
reversed and so transmitted to the valve, the position of the 
crank in relation to the eccentrics is indicated by the lines. 



49 

OK' (Fig. 27 and 28) as will be understood by consulting 
the rules given in the chapter on "the angle of advance." 

A right line drawn from the centre of the crank-shaft 
through the block 5 shall be termed the "centre-line " of the 
link motion. When the crank is on one of its centres, the 
centre lines of the eccentrics 1' and 1 " may form equal or un- 
equal angles with the line OF, which is drawn perpendic- 
ularly to OB. The reason why these angles are sometimes 
made unequal will be given later. 

The Theory. — The described link motion consists of a num- 
ber of oscillating and reciprocating parts, and the exact law 
of the movement which the link in its several grades of gear- 
ing transmits to the slide valve is extraordinarily complicated. 
Our theoretical investigation will therefore be based upon a 
series of assumptions similar to those we made when consid- 
ering the common slide valve. We shall thus disregard the 
effect of the oscillation or obliquity of the connecting rod and 
of the eccentric rods, and, besides, assume that every point of 
the link were moving in a right horizontal line, or, in other 
words, we shall overlook the vertical vibration of any part of 
the link motion. A subsequent chapter will then treat of the 
modification that our theoretical result suffers from these neg- 
lected influences, and show a method for finding the means to 
make these influences compensate each other. 

The mentioned assumptions would imply a distortion of the 
link during its movement, and to keep these distortions with- 
in as small a margin as possible, we shall study the move- 
ment of the link while in its mid-gear, when the link block 
will be operated by the centre of the link. It can then be 
demonstrated that the movement of any point of the link can 
be ccnsidered as though it were produced by a separate ec- 
centric, and it remains now for us to And the position of this 
imaginary eccentric. 



50 




51 

Suppose P and 1" (Fig. 29) to represent the two eccentrics 
of a link motion, the link P' P" of which is shown in its 
neutral position, obtained by transferring tne enf.s of the ec- 
centric rods to the centre of the crank shaft. Since we as- 
sume that all points of the link, and hence also the pins P f and 
P", were moving in horizontal lines, we should first under- 
stand the movement of the latter. Let us consider for this pur- 
pose the forward eccentric separately, in Fig. 30. 




Fig. 30. 

If the end P of the eccentric rod in this case were moving 
in the \\nep' p", the angular advance of the eccentric J for 
this movement would be the angle Y' 01, Y' being at right 
angles to OP. If it would be desired to produce the same 
movement on the horizontal centre line (by a horizontal ec- 
centric rod), the proper position of the eccentric would then 
be found at i by making its angle of advance YOi = Y'OL 
However, the movement of the pin Pis actually in the line 
P' P", and, therefore, when the eccentric 1 is at P or P f the 
end P will not be atp' ov p", but at P ; or P", respectively. 
The arcs p' P' and p" P" are very short in proportion to 
their radius, and may therefore be considered, approximately, 
to be right lines, forming right angles with P. Now it 
will be observed that the travel P' P" slightly exceeds p* p"> 
and hence also the diameter of the eccentric path Z, and to 



52 

produce a movement, on the horizontal line X, equivalent 
to that- of the pin P on the line P' P lf , we would have to in- 
crease the throw of the eccentric from i to J- by making J 
«= P P f . The point J can therefore be regarded the " ideal 
eccentric ' ' for the movement of P. 

A glance will now show that the angle 1 J is equal to the 
angle P X = [3, in other words, to obtain the ideal eccen- 
tric J, the eccentric 1 would have to be moved on the crankshaft, 
through the same angle and in the same direction cos the eccen- 
centric rod P would have to be moved to bring it in a horizontal 
position. Besides, the triangle 01 J is equal to the triangle 
Pp' P', whence it appears that the line J I is at right angles 
with 01. 

Applying these results to Fig. 29, we find the points J' and 
J" representing the ideal eccentrics for the movements of the 
extreme points P' and P" of the link, by making P J' = 
pi J" = ft and besides P J' and I" J" at right angles to P 
and 0P' y respectively. 

We are now prepared to study the movement of any point 
of the chord P'P f/ of the link, say of b. It can be asserted 
that, by dividing the line P J" in the same proportion in 
which the point b divides the chord P / P", a point j is ob- 
tained that can be considered the ideal eccentric of the move- 
ment of the point b. (The point b being on the forward end, 
P / V of the link, the point j?' is on the side of the forward eccen- 
tric J' of the line J / J n '. ) To prove this proposition we show, 
in Fig. 31, the crank K rotated through the angle a, and 
also the several real and ideal eccentrics, in their correspond- 
ing position. The link will then assume the position P / P /7 , 
and p / p" shows its neutral position, intersecting the line 
OX" in the point m. We then project, upon the horizontal 
centre line, the points P\ P ff and b and the ideal eccentrics 
J f P' and j, obtaining the points X\ X", X, x', %" and x. 

According to the foregoing demonstrations, the position 



53 



'fn» 



j?"i>' 



Fig. 31. 



of the points x' and x" in relation to will correspond with 
that of the points X' and X" in relation to m, and since the po- 
sition of x in relation to z' and x" will correspond with the 
position of X in relation to X / and X", on account of the 
proportional division of the lines J* J" and P / P // , it is evi- 
dent that the distances x and m X are equal ; or the hori- 
zontal movement of the point j equals the horizontal move- 
ment of the point 5, and our assertion is demonstrated. 

It must, however, be remembered that an eccentric axj can 
produce a movement as said only by means of a horizontal ec- 
centric rod, and if it was desired to produce the movement of 
the point b by a direct connection, it would be necessary that 
the eccentricrod should have an inclination ==/3 / (see Fig. 29). 
The case would then come under the same head as that con- 
sidered before (see Fig 30), and knowing the ideal eccentric 
j (Tig. 29) we can find the desired eccentric i by making the 
angles i = /?' and ji at right angles to Oi. Thus it is dem- 
onstrated that if an eccentric, at t, were added to the two ec- 
centrics F and /"; and were connected by an eccentric rod 
with the point b of the link these three eccentrics would ope- 
rate in perfect harmony ; but it should never be lost sight of, 
that this entire demonstration is based on our initial assump- 
tions and is therefore only approximately correct. 



54 






A lowering of the link that brings the point b in the centre 
line OX will also bring the imaginary eccentric i in a direct 
line of action, and the link block will move as if it was moved 
by this eccentric ; the point i is therefore the ideal eccentric 
of the movement of the valve for this grade of the link. 

The movement of other points of the link can be examined 
in the same way, and the result will be a series of " ideal ec- 
centrics," like % forming a continuous curve P IV* (Fig. 29) 
which may be termed " locus of ideal eccentrics/ ' or briefly 
locus, and it is demonstrable that this curve is a circular arc, 
tangent to the line J' J". 

Directing our attention to the quadrangle i°j i, we will 
notice that the angles at i° and at i are right angles, and 
therefore, if we draw the line ii°, the angles ii°j and iOf 
will be equal, according to a well known geometrical propo- 
sition. This shows that the line ii° forms right angles with 
the line Ob, since we made tiie angle i Oj = j3', which is the 
angla through "which the link and the eccentric rods have to 
be lowered (starting from the mid -gear) to bring the point b> 
and hence also the ideal eccentric i> in direct action. The 
same thing is true for every other ideal eccentric, and also for 
the real eccentrics P and P ! . 

Referring to this fact, we can find the centre of the "locus" 
by drawing a line through P, perpendicularly to P' , that 
intersects the centre line OX in i°, and by subsequently bi- 
secting the line P i° by a line parallel to P', that intersects 
the line X in the required centre o of the arc P i° P'. 

If now the circle of the locus is extended to its second inter- 
section at n, with the line OX, and we draw the lines nP 
and ni, it can easily be proven that the horizontal inclination 
of these lines will equal the angles /? and p f respectively ; 
hence nP is parallel to P f and ni parallel to b 0. This 
fict can very conveniently be used for locating on the locus 
the ideal eccentric for any point of the link. 



55 

In the foregoing we investigated the movement of a link 
with open rods. If, however, the rods are crossed, the in- 
clination of the eccentric rods will have the effect of diminish- 
ing, not increasing, the angular advance of the eccentrics. 
The ideal eccentrics for the movement of the ends P' and P ,r 
of the eccentric rods (Fig. 32) will be found to be located at 
J' and J n , in accordance with the rule found in relation to 
Fig. 30. The locus of ideal eccentrics can now be determined 
as before, and will again be found to be a circular arc, but its 
convex side will be turned towards the centre of the crank- 
shaft. The rules for finding the centre of this arc and for lo- 
cating the ideal eccentric for any point of the link, are the 
same for crossed as for open rods. 







If the link has the form shown in Fig. 26 it cannot be low-, 
ered or raised far enough to bring the pins P r and P" in the 
centre line, and the extreme positions to which the link can 
be lowered or raised were heretofore regarded intermediate 
grades. Only a portion of the link is available, and we can 
therefore use only a portion of the locus of eccentrics, as shown 
in Fig. 33. 

For convenience, the rule for constructing the locus of ideal- 
eccentrics shall here be recapitulated in a condensed form. 



56 



z 

/ 



i 
I 
i 

\ 

V 



\ 




jPtgr. S3. 

Make a skeleton drawing (Pig 29) showing the position of 
the eccentrics I' and I", relatively to the centre line of the 
link motion X, while the crank is on one of its dead centres 
Then draw the link P' P" in mid gear, in its neutral position 
(which may be done to reduced scale) and a line through the 
forward eccentric P, at right angles with the forward eccen 
trie rod OP', meeting the line OX in i® * Bisect the line P 
i° by a line parallel to P' and obtain the centre o of the 
locus* Sweep the locus I'i°I" and mark at the same time 
the point n of the same circle The ideal eccentric of any point 
of the link, say of b, is found by drawing through n a line 
parallel to the line b } which intersects the locus in the 
sought point i. 

The "Valve Diagram. 
According to the foregoing, the movement of the valve of 
the Stephenson link motion can be supposed to be derived from 
a series of eccentrics, and hence the distribution of the steam 
can be studied by a series of valve diagrams. To find them we 
perform the double operation of rotating the diagram of ideal 

* If the eccentric rods are crossed, the points P' and P" will ex 
change their places, and the centre oof the locu$ will accordingly 
be found on the other side of the crank -shaft. 



57 
eccentrics (l'L I") ihrpugb 90° and inverting it. This transfer 
or conveision can, in fact, most readily be accomplished by 
means of a piece of tracing paper, in the manner described 
on page 11. Thus we obtain the curve ty q° Q" (Fig. 34) on 
which the centres of the lap circles of the required valve dia- 
grams will be located. 




Fig. 34. 

The investigation of five different grades will fully suffice for 
a clear understanding of the. operation of the link, and the 
study of more intermediate grades will offer no difficulty. We 
shall choose for this purpose the lowest position of the link and. 
every following grade produced by successively raising the link 
one-quarter of its length, ending with the link in its highest 
position. The corresponding centres of the valve diagram (Fig. 
34) are Q', q', q°, q" andQ". Both lap circles can now be swept 
from each of these points, and we obtain for the first grade, 
forward, the crank angles OE, OC, OR and OA for expansion, 



58 
ousbioning, release and admission of steam. After drawing the 
erank circle Z', to reduced scale, we can find the corresponding 
positions of the piston, <?, c, r and a on the line 1, by projecting 
the points B, (7, B and A. The positions of the piston for 
the corresponding stages of the steam distribution of the other 
grades can be located in the same manner, and are marked on 
the lines 2, 3, etc. The crank-lines necessary for their con- 
struction are, however, omitted in Fig. 34, to avoid confusion. 
At all these grades the steam is admitted, on the steam side, 
until the piston arrives at e, it expands between e and r, and is 
exhausted for the rest of the stroke. The exhaust side of the 
cylinder is in communication with the exhaust passage before 
arriving at c, where cushioning begins, and at a the port is 
opened again and steam is admitted for the following stroke. 

Now, it will be seen that at grade 2, expansion, compression, 
release and admission take place earlier than at the first grade, 
showing that the link furnishes means for varying the degree of 
the cut-off. The action of the valve at the mid-gear, or grade 
3, is remarkably symmetrical as regards both ends of the cylinder. 
Expansion, compression and release begin at very early stages 
of the stroke, and the admission of steam for the following 
stroke, at a, takes place at such an early time that the piston 
has to force the admitted steam back into the boiler to complete 
Its stroke. This action will last as long as the admission of 
steam at the beginning of the 6troke, and the work then de- 
veloped will be entirely consumed. Steam is exhausted on the 
exhaust side of the cylinder from the beginning of the stroke 
up to c, where cushioning begins, w T hile on the other side steam 
is exhausted after the piston passes the point r. The steam 
expands while the piston moves from e to r, and the exhaust 
steam is compressed between c and «, which are again at equal 
distances, but as the mean pressure of the expanding steam 
generally exceeds the mean pressure at compression, there is a 
slight gain of work, which, however, is rarely sufficient to run 



; 



59 
the engine when without any load. This singular process is the 
same whether the crank-shaft rotates in the one or in the 
other direction. By reason of these facts the centre of the link 
is called, very properly, the '* dead point of the link." 

Passing over to grade 4 we find the action of the valve to be 
hastened still more, (see upper line.) The steam is now 
admitted on the exhaust side (after the piston passes the point 
■a) for a considerably longer period than on the steam side, 
(before the piston passes the point e) ; hence the motion of the 
crankshaft will be retarded — the admitted steam tends to rotate 
the shaft backward. At this grade the link is indeed raised so 
far that the backward eccentric I" exerts a predominating in- 
fluence upon the valve, and we might have arrived at the above 
•conclusion without the diagram. The distribution of steam 
will, therefore, take place as shown by the lower line. How- 
ever, the upper line, (grade 4) shows the process when a running 
engine is suddenly reversed. The engine will act like a pump, 
forcing air from the exhaust passage into the boiler until a 
stoppage and reversal of the rotation is effected. But as most 
locomotives exhaust into the smoke-stack, the air which is thus 
pumped into the boiler contains ashes in suspension which are 
apt to precipitate on the walls of the cylinder. It is, therefore, 
very injurious for the engine to reverse in this way, unless pro- 
vision is made to cut off the communication of the exhaust- 
passage with the smoke-stack. 

The line 5 (Fig. 34) shows the action, of the valve when the 
link is fully reversed. 

As regards the lead, the behavior of the Stephenson link- 
motion is very peculiar. The lead being represented by the 
distance of the lap-circle from the horizontal centre-line, the 
diagram (Fi<r. 34) indicates that the lead is variable at the dif- 
ferent grades. It is greatest at the intermediate grades and 
least at the full-stroke grades. It will further be Feen that the 
lead at the mid-gear is equal to the greatest port opening of the 



60 
same grade, the valve being at the end of Its travel when tha 
crank is on its centre. The greatest port opening is found to 
be greatest at the full-stroke grades and least at iirid-gear. 

J- - z ^ 




„ -7- 

Fig. 35. 

In Fig. 35 is shown the valve diagram of a liak-motion 
with crossed rods. We find in this case the behavior, relating 
to the lead, reversed, as compared with Fig. 34, and besides 
the greatest port openings for intermediate grades decidedly 
smaller. This latter is the principal reason why crossed rods 

are seldom in use. 

The Link and Its Suspension. 
The slot of the link should be curved in such a way that the 
gyp* ' 'centre of movement" or ' ' neutral 

ij ****^ position" of the valve should be the 

1 1 ^^ same for every grade of the link- 

il 1 y*}F Hence, if we transfer the ends of 

the eccentric rods to the centre 
(Fig. 36) of the crank- shaft a lower- 
ing or raising of the link should not 
affect the valve. This condition is 
fulfilled when the curvature of the 
link (in the position shown in Fig. 
36) is a circular arc swept from the 
centre of the crank-shaft. Both 
Fig. 36. eccentric rods should, therefore* 

be of equal length, and the radius of the centre line of the slot 




61 
-equal to the length of the eccentric rod plus the distance of 
the pin-hoie P' (Fig. 25) from the centre line of the slot. If 
the link is as shown in Fig. 26, the latter distance is = zero. 

The link should be made neither too long nor too short. A 
•definite rule cannot be given to comply with the different con- 
ditions to which link motions may have to be adapted. Good 
results may be obtained, in most cases, by making the available 
length of the slot of the link about six times the throw (Oi') 
of the eccentrics. The actual length, of course, must be longer 
by something more than the height of the link-block. 

The mode of suspension of the link has a decided influence 
upon the regularity of the action of the link-motion. In our 
investigation we assumed that every point of the link were 
moving'in a horizontal line. To approach this supposition as 
near as possible, the suspension rod GD (Fig. 36) should be 
made as long as space and convenience will permit, in order to 
flatten the arc in which the point D moves. Besides, to keep 
this arc in a horizontal position, the neutral position of the rod 
GD should invariably be vertical, whether the link be raised or 
lowered, and this will be the case if the upper point G of that 
rod is guided in an arc G' G", which is equal to the arc D / D" y 
in which the pin D of the link is moving as the link, in its 
neutral position, is lowered and raised. The centre F of the 
suspensiou lever should, therefore, be vertically above 0, at a 
distance equal to the length of the hanger GD, while its length 
FG, should be equal to the distance OD in Fig. 36, 

If the link is suspended by the lower pin P", Fig. 37 will 
have to be substituted for Fig. 36, in the above demonstration. 
In practice, the length of the suspension lever, FG, is seldom 
made of the length required by this rule, sometimes for want 
-of space, at other times on account of the inconvenience of 
such an extreme length. Besides, it is a general practice to 
purposely introduce certain irregularities by the mode of 
suspending the link, to compensate for the unequality between. 



62 
fore and return stroke, caused by the angularity of the connect- 
ing rod. This matter will be discussed later. 




Equalization of the Lead. 

We have seen that the^lead of the Stephenson link motion iff 
"Variable with the different grades, and this is regarded by some 
engineers as a disadvantage of this valve gear. It is, however, 
questionable if this objection is really worth much attention. 
We shall, therefore, but briefly discuss the method that is 
sometimes employed to equalize the lead. 

The said variation of the lead is owing to the change of the 
inclination of the eccentric rods in changing the grade, and 
will, therefore, be reduced by the use of long eccentric rods 
and a comparatively short link. It can be traced back to the 
variation of the locus l'i° /"(Fig. 29) from a vertical line. 

The method alluded to consists in giving to the forward 
eccentric V a greater angle of advance than to the backward 
eccentric I", (Fig. 38), when the locus of ideal eccentrics can 



• rtl 




63 
be found as before, the result 
being the arc l'i°l". The lower 
half of this curve now approaches 
more nearly a vertical line, (com- 
pare Fig. 38 with Fig. 29), show- 
ing that the lead will be less 
variable at the different forward 
grades, and this variation is least 
if the point i° is vertically above 
1\ The upper half of the locus 
in Fig. 38, however, deviates con- 
siderably more from a vertical line than before, hence the 
equalization of the lead for the forward grades is accomplished 
at the expense of the lead of the backward grades. 

A link-motion with equal angles of advance can be changed 
to have equalized lead at the forward grades by advancing- 
both eccentrics through a certain angle, in the direction of the 

for ward rotation of the shaft, 
or else by moving the crank 
in the opposite direction 
through the same angle. In 
this latter case, nothing in 
relation to the link-motiou 
is disturbed, and the original 
diagram of ideal eccentrics 
(Fig. 39) will serve, the only 
difference being a change of 
the crank from OK to OK' y 
and the latter line is the base line for the new diagram, as 
regards the angles of advance, while OK remains the centre 
line of the locus. This result is identical with that obtained 
before in Fig. 38, but by this way of reasoning we can* easily 
ascertain through what angle the crank should be shifted in 
order to get the line l'i° at right angles with the new base line 



r 




64 
0K\ This angle is evidently = I'i°J"=p. (See Fig. 29.) 
If the link is as shown in Fig. 40, the crank need only "be 
shifted so that the points i' and i° (Fig. 33) are in a vertical line, 
and hence the angle ji' (Fig. 40) should be used for the cor- 
rection. 




Fig. 40. 

The corresponding valve diagram is shown in Fig. 41, where 
it will be seen that the lead remains nearly the same for the 
grades Q' q' and q°, but it diminishes rapidly, as the link is 
raised farther and adjusted to the grades q" and Q". At the 
latter grade the lead may even become negative, in other words, 
the opening of the steam-port takes place after the crank passed 
its centre. 

This method can evi- 
dently be applied with 
advantage only on en- 
gines which are princi- 
pally used running for- 
ward, as is the case with 
marine e gines and loco- 
motives. 




Go 



Problems. 

Problem VIII. — A Stephenson link-motion, with open rods, is 
to be constructed, to cut off at ^ of the stroke when at full 
grades, with a lead of J". The largest port opening for the 
full grades is required to amount to If" and the inside lap to be 
J". The link is of the form shown in Fig. 25, its length P'P" 
is 12", the distance of the pins P / and P ;/ from the centre-line 
of the slot is 3" and the length of the eccentric rods is 3' 6". 
The link-block is directly geared to the valve- stem, and at full 
grades the eccentric rods are to be in line with the link-block. 
The proper dimensions and positions of the valve and eccentrics 
are required, and besides the elevation of the link when required 
to cut off at J- and J- of the stroke. 

Since we know the greatest port opening, the lead, the point of 
cut off and the inside lap, for both full grades, we can find the 
points Q' and Q" of the valve diagram (Fig. 42). Next we 




Fig. 42. 

locate the link P'P" (Fig. 43) in its neutral position in mid 
gear at reduced scale. Mark the eccentrics P and I" to correspond 
with Qf and Q" of the valve diagram, draw the line Pi° at right 
angles to P'O and bisect Pi? parallel to PO to get the centre o 
of the locus. The locus can now be drawn in Fig. 43 and trans- 
ferred to Fig. 42. In Fig. 43 the point n can also be marked, 
either by extending the circle of the locus, or by drawing Pn 



66 
parallel to OP'. To find the positions of the link for the re- 
quired intermediate grades we draw the crank angles OE' and 
OE" (Fig. 42) to correspond with f and f of the stroke of the 
piston, and find on the curve Q'Q" the centres q' and q" of 
the lap circles that are tangential to these crank angles. Finally 
we transfer the points q f and q" to Fig. 43 (i* and i") and draw 




ti 



Fig. 43. 

the lines Ob' and Ob" parallel to i'n and i"n, and b' and b" are 
the points of the link for the required intermediate forward 
grades; those for the corresponding backward grades are in a 
symmetrical position. We fmd,besides the angle of advance =<J, 
the throw of the eccentrics 01' =01" = 2%" the outside lap 
= 1" The radius for the centre-line of the slot of the link 
should be =3'6"-f3"=3'9". 

Problem IX. A Stephenson link-motion with open rods is 
required with a link of the form shown in Fig. 26. The cut- 
ting off for the first forward grade is to take place at £ of the 
stroke, with a lead of \". The largest port opening is to be 
== If" and the inside lap \". It is desired that the lead should 
be equalized for the forward grades. The length of the eccen- 
tric rod is 3'6", the link-block is attached, at a leverage of 9", 
to a rocker, from which the motion is transmitted to the valve 
by an arm of 12" length. The centre line of the* engine is in- 
clined to the centre-line of the link-moton, by an angle of 15°, 



67 
as shown in Fig. 44. The length of that portion of the link (b'b" 
Fig. 40) that is actually used is to be equal to three times the 
travel of the link-block when in full gear. It is required to con- 
struct the valve diagram, to find the throw and the position of 




Fig. 44. 

the eccentrics, and the length of the link shall be determined, 
if the distance / 'b' (Fig. 40) is=3". Any question relating to 
intermed : ate grades can, of coarse, be solved as in the preceed- 
ing problem. 

The required valve diagram is shown in Fig. 41. At first we 
can construct the centre Q' for the first forward grade, showing 
the travel of the va've at that grade to be 4^" with an outside 
lap of |". Owing to the unequal leverage of the rocker, the 
corresponding travel of the link block is 3f " and the length b'b " 
of the link is accordingly 10 J". The length of the eccentric 
rods 1 eing given, we can now locate these two points of the 
link, at reduced scale, (Fig. 45) and find the angle /3', which is 
the angle of equalization of the lead. We can thus consider 
the casein the way shown in Fig. 39, i.e. — we consider the line 
Ob' (Fig. 45) as the base line of the diagram of eccentrics, and 
can accordingly transfer the point Q' (Fig. 41) of the valve 



68 
diagram to its proper position V in tiie diagram of eccentrics 
Fig. 45, by ihe angle 6. The locus can next be drawn, as usually, 
and it should be extended to bo:h sides, in this case. The 
drawing of t'le link can then be completed by making &'P'= 
b n P"=3", and the points J' and i", corresponding to the points 







Fig. 45. 



P' and P" are found by making nP and ^"parallel to P'O and 
P"0 t respectively. But for the unequal leverage of the rocker 
the points P and 1" would represent the eccentrics by which the 
points P f and P ff of the link should be worked, and their throw 
would be 3£". But as the case is, this figure is subject to a re- 
duction and the actual throw of the eccentrics must be made 

2 T V" 

When we complete the valve diagram, Fig. 41, we find the 
lead for the first backward grade to be negative, about \" . which 
is certainly not a very desirable feature. The variation of the 
lead in the present link-motion assumes such proportions, on 
account of the rather unusual shortness of the eccentric rods. 






69 

The slot of the link should be curved by a radius of 3'G", 
equal to the length of the 
eccentric rods. 

To find the proper posi- 
tion of the eccentrics on K / \ I \\ 

the crank shaft we deter- '"'■ \{\ — ' 

mine the angles c5' and 6'\ — - -r — r^**A — j 

the line O J 7 forming right \ h ' 

angles with the base line \ r. / 

VO (Fig. 45). Then we \^ lis* / 

draw the crank OK (Fig. **!*££ -^ 

46) on its centre, which is 1 

at an inclination of 15°, Fig. 46* 

and locate the eccentrics l f and 7" by their angles of advance, 

6' and 6". 



Irregularities and Their Rectification. 
The results of tbe foregoing considerations will be somewhat 
modified when we take into account the complicating elements, 
of the link and crank motion. 

The effect of the angularity of the connecting rod has been 
considered before. 

The irregularties arising from the oscillations of the eccentric 
rods can be corrected by adjusting the valve to have equal lead 
at the fore and return stroke. 

The oscillation of the link has the effect of drawing the ends of 
of the eccentric rods nearer the centre line, and since this 
(by reason of the inclined position of the eccentric rods) is accom- 
panied by a horizontal movement from the crank-shaft, this 
influence is equivalent to a temporary lengthening of the eccen- 
tric rods while the link is inclined. 

Another source of variation from our theoretical results is the 
"slip." Owing to both the mode of suspension and the oscillation 
of the link, the latter is caused to slip up and down on the 
block, and the valve is not operated from a fixed poirt of the 



70 
link. The sUp depends mainly on the mode of suspension and, 
as a general rule, has likewise an effect equivalent to a tempo- 
rary lengthening of the eccentric-rods as the link assumes an 
inclined position. The slip can be reduced by lengthening the 
link and by suspending it as shown in Fig. 36 or 37. 

There are still other influences of a similar character, but 
being of minor importance their special mention is here omitted. 

It will be remembered that the cutting off on both ends of 
the cylinder can be rectified by lengthening the eccentric rod, 
but that this proceeding will render the lead unequal for both 
strokes. Hence, if we could temporarily lengthen the eccentric 
rods for the time of the cutting-off and restore their original 
length for the beginning of each stroke, the cutting-off could 
be equalized without destroying the equality of the lead. Now, 
w r e have seen that the link-motion possesses a feature of this 
description, and by a judicious choice of certain dimensions we 
may correct one irregularity by the other. This correction, it 
will be seen hereafter, is accompanied by rn increase of the 
slip, anclwe have to choose one of two evils: increased slip, 
which is attended with greater wear on the link-block, or un- 
equal cut-off, which is attended with a less uniform motion. 
The former is preferred by most practical engineers. 

We may consider the subject of rectification in a somewhat 
different light. It is known that the cylinder takes more steam 
on the fore-stroke than it does on the return-stroke, if no recti- 
fication is attempted. „ Remembering, now, that with the link- 
motion we have the power to change at pleasure the de- 
gree of expansion, it is easily seen that a raising of the link on 
the fore-stroke andaloweiing on the return-stroke does pre- 
cisely what w T e want. There are two means at ( ur disposal 
that will accomplish this automatically. We may shift the 
suspension pin of the link, laterally, towards the inside or out- 
side of the link, as the case may require, and employ the 
" oscillating movement " of the link for the purpose; or we 



71 
may place the hanger in an inclined position, turning the 
"reciprocating movement" to our use. 

Attention should, however, be paid to one circumstance. If 
the link is suspended by its centre, there are two grades (one 
forward and one backward grade) at which the suspension pin 
of the link, as the valve is cutting-ofT at the return-stroke, will 
have returned to the same position it occupied at the same 
period of the fore-stroke, and which we may call, for future 
reference, the " reciprocal grades." At these grades the cutting- 
off is effected the moment the imaginary ecentric i° (Fig. 47) 



Fig 47. 
of the movement of the suspension pin D is at right angles to 
its imaginary line of operation, OD. The inclination of the 
hanger can, therefore, not produce the required lowering or 
raising of the link at either stroke, and we can employ only the 
oscillating movement of the link for correction. 

Since we are restricted to guide the upper end of the hanger 
in a circular arc, the radius of which is limited to a certain de- 
gree, we cannot, alter correcting the reciprocal grades, accom- 
plish the perfect rectification of any more than two additional 
grades, and it is best to choose the full stroke grades for this 
purpose. The other intermediate grades will then be found so 
nearly rectified that there will be no need for any further im- 
provement. 

The number of grades for wirich a correction is possible, when 
the link is suspended on its lower end, is likewise limited to 
four, and in order to bestow the rectification upon advan- 



72 
tageously distributed grades, a careful lateral location of the* 
suspension pin is necessary. If a choice in this respect is ex- 
cluded, for instance, if one of the pins of the eccentric rods is 
used for suspension, we must be contented with a less perfect 
rectification, or as the case may be by bestowing a perfect rec- 
tification upon the forward grades only. 

The selection of the grades for perfect rectification of the- 
cut-off must be left to the judgment of the designer; it is,, 
however, not necessary to take any of those grades for which 
the arc of the reversing-lever is notched, as any other interme- 
diate grade is equally serviceable. As a rule, the full stroke 
grades, and those for cutting-off at half stroke, for both forward 
and backward rotation, are a fair base for correction, and they 
have been chosen in the following practical examples. 

After these preliminary remarks we may take up the discus- 
sion of the way for finding the proper dimension of the sus- 
pension mechanism. 




Fig 48. 



I i T' 



a 






I I 



\\ 



I I 
/ i 
I I 



I i 



Nee 



Fig. 49. 



The complication and variety of the disturbing influences; 



73 

j-ender it almost impossible lo biing them fully to account 
without the construction of an actual model, or an equivalent 
thereof. Such a model can be prepared of stiff drawiug-paper 
or thin wood, aud attached, by thumb tacks, to the drawing- 
sheet on which the suspension lever is to be laid out. It should 
always be made to full size to insure sufficient precision. 

The principal dimensions of the hanger, the link, the eccen- 
tric rods, the eccentrics, e l c , should be determined to suit the 
special case, and, as shown in the two problems, the first of 
which shall be used for exemplification, supposing the connect- 
ing rod to be equal 5 cranks, and the hanger =16". 

The link may be made as shown in Fig. 48 or 49, and should 
be cut carefully along the cent re-line of the slot. The eccen- 
tric rods may be made of two strips of paper and provided 
with two pin-holes at their ends, at the exact length of the con- 
templated eccentric rods, and they should be marked *f oward" 
and "backward," or "+" and " — ", for contradistinction. 

We now draw the centre line of the link-motion (Fig. 50) and 



a'c 







2fc3 



- /, Z 



I -^— 



4-' 
2 £3' 



' ^-*1 



Fig. 50. 

sweep from O the orbit Z of eccentrics. Then we construct 
the crank-angles, at which the cutting-off of tie four grades 
that we selected for the base of correction (which are^ andj 
of the stroke), takes place, by the proper circular projection. 
In Fig. 50 these crank-angles are marked 1, 2, 3, 4, and 1', 2' % 



74 
3', 4', the former standing for the fore-strokes and the latter 
for the return -strokes. 

The link can now be attached to suit any crank-angle, by finding,, 
by means of the angles of advance, the corresponding positions 
of the two eccentrics, and pinning the eccentric rods to those 
points, care being taken not to pin the '* forward rod" to the 
"backward eccentric." The free ends of the rols may be 
fastened to the link by thumb tacks, with their points up. 

The mode of procedure is now as follows : Attach the link 
to the first dead centre of the crank K, as shown in Fig. 50 by 
dotted lines, pull slightly in the direction of the darts, to stretch 
the paper straight, set the link to mid-gear, and mark with a. 
sharply pointed pencil the point a. Repeat this operation for 
the second dead centre of the crank, K', and obUin the point a ^ 
Bisect the distance a a' in c and regard this point the centre of 
movement of the link-block. In this way we secure an equal 
lead as regards fore and return stroke at the mid gear. If the 
link-block is attached to a rocker, let the arc of the movement 
of the block pass through the points a and a'. Then mark 
the positions of the block, b and b', at which the valve will just 
close the ports by making cb—cb' equal to the outside lap of 
the valve (du'y r di;ced, of course, when a rocker of unequal 
leverage is interposed). 

Now, suppose the link was to be suspended at its lower end,, 
somewhere in the line n (Figs. 48 and 50). Attach the link suc- 
cessively to all the crank-angles marked before, adjust its curved 
edge to b or b' y accordingly as the crank-angle pertains to the 
fore or the return stroke, and mark the pos.t'on of the line nof 
the link and its termination (or intersection with the link-curve) 
on the drawing. We thus obtain the eight lines shown in Fig. 
51, where they are marked by the same figures as the corres 
ponding crank-angles. They represent, as it were, the positions 
the link will occupy at the respective periods. 

We have now to find the lateral location of the suspension 



75 
pin on the line n of the link. To this end we first suppose it 
to be located in the centre-line 
of the slot, or, in other words, 
in the termination of the Ymen. 
We sweep accordingly from the 
end points of the lines 1 and V 
two arcs of a radius of 16" (the 
length of the hanger). Their 
intersection gives the point G° 
from which the hanger should 
be suspended for the first grade 
The points of suspension, GG, 
for the three other grades can 
be found in the same manner 
bat it will be seen that no one 
circle can be drawn through 
these four points, hence they 
do not satisfy the required con- 
ditions. However, if we move 
the suspension pin of the link 
laterally, we can find, after a 
few attempts, a point, marked 
D on every one of the lines of 
Fig. 51 that furnishes points of 
suspension, g, g, which can be 
connected by a circular arc. 
The centre, F, of this arc deter- 
mines the location of the ful- 
crum, and its radius the length 
of the required suspension lever. 
The point D can now eas'.ly be transferred tothelink, and the 
designer can accordingly finish the detail drawings. 

The above course of procedure will have to suffer a slight 
modification if the link is to be suspended in its centre-line m 




Fig. 51. 



G fr « 




76 
(Fig. 48). The eight different locations of the link must be 

marked by the centre line ra, 
and will appear as shown in Fig. 
52. It will be observed that the 
lines 2 and 2', as well as 3 and 
3', intersect in a p )int? from which 
the end points of these lines are 
nearly equidistant, simply because 
the grades 2 and 3 are in close 
proximity to the reciprocal grades. 
By rights we ought to have taken 
those latter grades instead of 2 
and 3, but we obtain practically 
the same result by averaging the 
distances of the Slid points of 
-7-p^-.^ intersection from the termination 

,,, , / *? of the respective lines aud take 

^?L.L./i.-.-o. -./-.-i« this distance for the lateral posi_ 
/ tion of the suspension pin D 

1 which then cau be located on the 

^f>j?^ / eight liak-lines. Next we deter- 

/ mine the points of suspension, 67' 

.^o^^ / and 67", for the grades 1 and 4, 

i)^^* and mark through th.se points 

L < ^ / the arc of the suspension lever. 

I The latter should be made as 

/ long as convenient, in order to 

pn I have the intermediate grades more 

© I nearly corrected. 

When a link-motion is rectified 
i in this manner as regards the cut- 

-j ting-off, it will be found to be 

nearly so for the cushioning and 
Fia. 52. ttie release - 0ne of tne a S enc i es 



1 






77 
employed for the rectification, namely, the inclination of the 
hanger, will slightly affect the lead, especially at the full-stroke 
grades, but the difference is so small that it can hardly be noticed. 
The port-opening, indicated for any crank-angle by the diagram, 
will also slightly deviate from the actual port-opening of the link~ 
motion, and will generally differ at the corresponding periods 
of both strokes. The greatest port-opening may thus become 
smaller than premised. 

It must not be overlooked that the rectification is accom- 
plished at the expense of the slip, and it is therefore necessary 
to sufficiently lengthen the slot of the link for this purpose. 
The paper model affords the means for finding the extent of 
this lengthening. 

The slip may sometimes become excessive, and it may then 
be expedient to abandon the perfect correction and compromise 
between this correction and the slip, by suspending the hanger 
more nearly in a vertical line than demanded by the above rule. 
Or the valve can be made with unequal lap for both ports, cor- 
recting the cut-off at the expense of an equal lead. 

The described manner of attaching the eccentric rods and 
the link by thumb tacks may involve some inaccuracy. Greater 
precision maybe attained by drawing the link on tracing-paper, 
.and instead of using the paper eccentric rods, sweeping arcs 
from the respective positions of the eccentrics, and by adjust- 
ing the pin-centres of the link to ihose arcs, the position of the 
link can be obtained as before.* 

* This latter method for findiDg the position of the link for any given 
crank-angle has been used in England. Auchincloss, in his treatise on 
., Link-motion," recommends the use of a template cut of thin wood, to be 
adjusted to those arcs. 



78 

The Gooch Link-Motion. 
This link-motion was invented independently of and almost 
simultaneously with the Stephenson link-motion, and is dis- 
tinguished from the latter principally in that the link is sus- 
pended at a stationary night, the link-block being moved in- 
stead, by the reversing lever, as illustrated in Fig. 53. The 




Fig. S3. 

connection between the link-block and the valve-stem is effected 
by the radius rod BP. 

To investigate the theory of this link-motion, let us again 
neglect the influences of the oscillation of the several members, 
and consider the movement of the pins P' and P" to be strictly 
horizontal. The centre line of this link-motion is the line 
passing through the crank-shaft O and the pin P of the valve 
stem. The movement of the chord P'P" of the link is now 
precisely the same as that of the shifting link when in mid-gear. 
Supposing that V and l ff (Fig. 54) were the eccentrics, and 
P'P" the link in its neutral position, we can find the ideal ec- 
centrics J"' and J" of the movements of the pins f ' and P", by 
moving the real eccentrics 1' and I" through the same angle (/5) 
and in the same direction as the corresponding eccentric rods 
would have to be moved to make them horizontal, and then 



79 
drawing the lines I'«7' aad I" J" at right angles to OF and OP'. 

The horizontal movement of any intermediate point, 5, of the 
chord of the link can again be shown to be equal to the hori- 
zontal movement of a corresponding pointy of the line J' J", 
in the same way as before (see Fig. 31); aucl siuce in this link- 
motion the movement of this point is directly transmitted to 
the valve when the link block is raised to the point B (the move- 
ment of the points B and b being practically equivalent,) the 
point j is the ideal eccentric of the movement of the valve, 
and the ideal eccentric for any other grade is situated in the 
straight line J' J", dividing it in the same proportion as the 
position of the link-block divides the chord of the link. The 
relation existing between the points b and J can again be estab- 
lished by parallel lines. Draw a line through J' parallel to 



-d-r. 




Fig. 54; 

O P' and extend it to its junction, at n, with the horizontal 
centreline. The line joinings' and n will then be parallel to 
that joining O and b. The valve diagram, Fig. 55, can now 
readily be drawn, showing the lead of this link-motion to be 
equal throughout all grades. In other respects, the feature of 



80 
the variableness of the expansion and the exhaust is very 
much like that of the Stephenson link-motion. 

The curvature of the link, having its c< nvex side turned 
towards the crank-shaft O, should have a radius equal to the 
length of the rod BP (Fig. 53) as will be seen by setting the 
link in its neutral position, where a lowering or raising of the 
link-block must not produce a movement of the valve. The 
link is mostly suspended, having its centre in the centre line. 

If crossed eccentric rods are employed, the eccentrics should 
have the positions i f and i" (Fig. 54), when the corresponding 
eccentrics will again be lepresented by J' and J", and the dia- 
gram of the ideal eccentrics will be identical with the one for 
open rods. 



+ 




Fig. 55. 

To reduce the slip to a minimum both suspension rods Gl> 
and G'D' (Fig. 53) should be vertical when in their neutra 
position. To this end the point G should be guided in an arc 
equivalent to the arc made by the point D when the link is in 
its neutral position. The fulcrum F should, therefore, be ver- 
tically above the neutral position of the pin P, at a height 
equal to the length of the hanger, 6ri>; and the length of the 
lever FG should be equal to the length PD. In practice, how- 
ever, this lever is usually made shorter, and is located to effect 
a rectification of the cutting off at fore and return strokes. 

The advantage of this link-motion over the one treated before 
consists in the permanency of the lead throughout all grades* 



81 
while the increased number of parts, the less length that can be 
given to the eccentric rods at a given distance of the crank shift 
from the valve-chest, and the greater side pressure that the 
valve-stem receives from the inclined rod, involving the neces- 
sity of a guide for the valve-stem, are doubtless disadvantages. 

Problem X. — A Gooch link-motion shall be designed to cut 
off, when in full gear, at three quarter stroke, and to cushion 
for one-twelfth of the stroke. The greatest port-opening is 
required to be 1£" with a lead of £'■ . The link is 12" long and 
the eccentric rods are 2' lj" f rom centre to centre. It is re- 
quired to draw the valve diagram and to find the throw and 
position of the eccentrics for both open and crossed rods. 

The centres Q J and Q" (Fig. 55,) for the full gears can again 
be lound as usual, the construction yielding an outside lap =1", 
an inside lap = 1 3 g- // and the travel of the valve =4| / »\ To Snd 
the required throw and position of the eccentrics we draw the 
link in its neutral position (see Fig. 54), to find the angle /?, 
carry the same to 'joth sides of the lines OJ' and OJ" (J 7 and J' 
to correspond with the points Q' and Q' of the valve diagram, 
Fig. 55), and draw the lines J'/', J'i'\ J"I" and J"i" at right 
angles to OP Oi',OP f Oi", respectively. The points P aud P / 
represent the required eccentrics if open eccentric rods are 
used, and i* and i" those for crossed rods. The throw of the 
eccentrics can be measured by the lines OP, etc., and is found 
=%fe '• The angular positions of the eccentrics are given either 
by the angles d'or d". Both eccentric will then have an angle of 
advance =d in relation to their eccentric rods. 

The proper points of the link producing certain grades of 
cut-off can easily be found by the use of the valve diagram. 

A rectification of the cut-off of four grades can be accom- 
plished, as in the Stephenson link-motion, through the oscillation 
of the link and the movement of the hanger GD. The sta § 
tionary hanger *G'D' cannot be made available for this purpose, 
ani should, as a rule, be suspended vertically, in its neutral 



82 
position, except if the foward grades are to be favored at the- 
expense of ihe backward grades ; for the rectification of the 
backward grades will call for exactly the opposite inclination 
as the rectification of the fovard grades. A paper model (see 
Fig. 56,) can again be used to find the suitable position of the 






U 91 







\ i db'ij: ba . 




Fig. 56. 

stud B and the suspension lever FO. After drawing the orbit 
of the eccentrics, and the crank-angles for the cutting- off of the 
four grades, we desire to equalize (compare Fig. 50), we attach 



83 
the link to correspond with the dead centres of the crank, 
mark the points a and a' (Fig. 56), bisect their distance in 
c and mark the points b and b' representing the position of 
the link-block (ii mid-gear) when the cutting-off takes place. 
Then we sweep the paths, through b and b', of the link-block 
when raised or lowered for the said positions of the valve. 
^Next we assume, at pleasure, a point of the link, say I) f (Fig. 
56) for the suspension pin, and make bd=b'd'=mD', in order 
to locate the suspension fulcrum G' by sweeping two arcs from 
d aud d', of a radius equal to the length of the hanger G'D'. 
This being done we set the link for the crank-angle 1, adjusting 
the ' point D' of the link to the proper distance from G' and 
mark the intersection 1 of the link-curve with the arc b. We 
repeat this for the crank-angle 1', marking the point 1/ on the line 
b', and continue with the other crank-angles to obtain the points 
2, 2', etc. Draw the corresponding positions of the valve-rod 
1^?, l'jo', etc. ( the points^? and p', towards which the two sets 
of lines converge, are not shown on the drawing), and mark 
the points D,D, of the suspension pin of that rod. Two arcs 
iswept from the points -D,Z>, of the lines lp, and l'p' locate by 
intersection the point G° from which the hanger of the valve- 
rod should be suspended for having the first grade rectified. 
The points of suspension G,G,G, for the other grades will be 
found by the same method. 

But, whereas, these four points are not located so that one 
circular arc can pass through them, they do not suit our pur- 
pose. We have, therefore, to change the assumed position of 
the suspension pin D' of the link laterally, and to repeat the 
above operation until the suspension points can be connected by 
a circular arc, which then determines the length and position 
of the suspension lever. 

In the case shown in Fig. 56 the suspension pin D would 
nave to be on the other side of the centre line of the link to 
obtain this result and the location of the suspension lever would 
be very awkward. The stud D may, therefore, be located in the 



84 
centre line of the link, when the points^ will be found and 
the suspension lever may be made to pass in their close prox- 
imity, as indicated by a dotted arc. It will now be noticed that 
this arc passes exactly through the points of suspension for the 
backward grades, but is less favorable for the forward grades. 
Since the reverse would be the more desirable, we may reverse 
the whole arrangement, i.e., locate the suspension lever below 
the centre line of the link-molion. Or, if a reduction of the 
slip is more desirable, the valve may be made with unequal lap 
when the correction of the cutting- off can be convenienily ac- 
complished with a minimum amount of slip, at the expense of 
the equality of the lead. Unequal lead is of very little conse- 
quence,because the position of the piston for the moment of the 
opening of the port will be scarcely affected by a slight change 
of the lead. 

Another expedient, sometimes adopted to better equalize the 
distribution of steam for the forward grades, at the expense of 
the backward grades, consists in suspending the link so that 
its centre is below (or above, as the case may require) the cen- 
tre line of the link-motion. 



PART III. 

INDEPENDENT CUT-OFF GEARS. 



87 



Independent Cut-Off Gears. 

As perfect as the link-motions may appear as regards the 
variableness of the cutting off, they notwithstanding possess 
one undesirable feature. They admit of a regulation of the 
admission of steam, but the exhaust is dependant on this regu- 
lation, and the objection existing against the use of an early 
cut-off by the common slide valve, holds good for link-motions* 
For this reason the " independent cut-off valves" have been 
devised, which are combinations of two or more valves that 
admit a variation of the cut-off independently of the exhaust. 
They are very extensively in use on larger stationary and marine 
engines, but their applications to locomotives did not meet with 
much favor on account of their complication. 

The functions of admission and of exhaust may be entirely 
separated, i. e. y distinct valves may be used for each of those 
functions : or a main- valve similar to that of the common slide 




wmmmm 



Fig. 57. 

can be made to control both admission and exhaust, leaving to 
the second or cut-off valve the sole duty of closing the admis- 
sion of the steam at an earlier period than the main-valve does. 



88 

The first class includes the disengagement valve gears, which 
are generally of a more complicated construction, although 
their theory is very simple. Gears of the second class are less 
complicated, but their study offers more difficulty, and this 
class will be considered inihe following • 

Such valves may be arranged in two different ways. In 
Pig. 57 a partition divides the interior of the valve-chest in two 
chambers, in each of which one of the valves are placed, 
which are moved independently of each other. Both valves 
are shown in their neutral position. The cut-off valve V°, is a 
plate with rectangular perforations or ports coinciding with 
equal perforations of the partition. A movement of this valve 
exceeding the width of these ports will interrupt the admission 
of steam. One single opening in both the valve and the parti- 
tition valve would answer, but it is preferred to make a number 
of smaller ports, in order to reduce the travel without reducing 
the total port opening, for the sake cf reducing the loss of 
power from friction. Such valves are inown by the name of 
" Gridiron valves." 

The main valve Vis of the same construction as the common 
slide valve. 

The second way is shown in Fig. 58. The cut-off valve V° 
is placed directly over the main-valve V, and to make it ef- 
fective, the main valve is extended on both sides to form two 






jBl. 



ff=f\ 



ff=ti 



rPti 



i%%0%ss0a30g^^ 




wmmmmmtv 



Fig. 58. 



89 
steam passages, v and v'. These valves are again shown in, 
their neutral position, to show their relative proportion and 
their laps ; they will, however, rarely attain this position sim- 
ultaneously when in action. 

These valves are usually operated by two separate eccentric?. 
The action of the main valve V is identical with that of the 
common slide valve ; the cut-off valve V°, in closing the 
passages of the main valve, will interrupt the admission of 
steam earlier than the main valve will. The operation of this 
cut off valve requires a special study; for the present case 
differs from our former investigations in that this valve is not 
sliding over a fixed seat, the seat being formed by the main 
valve, which has a movement of itself. What we have to 
consider is not the actual movement of the cut-off valve, but 
its movement in relation to that of the main valve. 

Turning our attention to the position of the eccentrics / and 
1° (the latter being the cut-off eccentric) in relation to the 
crank O iT(Fig. 59), we note the movement of the main valve 
to be represented by the distance t 1 where t is the horizontal 
projection of the eccentric I. The absolute movement of the 
cut-off valve is the distance of the projection t° of the cut-off 
eccentric 1° from the centre O, but this distance is of little 
value for our investigation. The relative movement of the 
cutoff valve, i. e., the distance of the centre-line of the cut- 
off valve from the centre-line of its seat, the main valve, is the 
distance 1 1°, and this distance is the horizontal projection of 
the line 1 1°. 

If we now draw the line J° parallel to 1 1°, or in other 
words, if we shift the line I 1° parallel with itself along the 
centre line I of the main eccentric to O J°, it is evident that 
the projection t' of the line O J° must be equal to the pro- 
jeciion 1 1° of the line I 1°, which is equal to the relative- 
movement of the ciU-off valve ; and if we can prove this rela- 
tion of the line O J° to remain unimpaired by a rotation of the 



90 



•crank, then the point J° must be regarded as the ideal eccentric 
of this movement 








Fig. 60. 



To prove this relation, let us view the case after the crank is 
rotated through the angle a (Fig. 60). The relative movement of 
the cut-off valve (which, by the way, is negative in Fig. 60), is 
again represented by the distance 1 t°, which is likewise equal to 
the projection V of the line J°. 

The relative movement of the cut-off valve is therefore equiva- 
lent to the movement of the same valve over a fixed seat when " 
operated by the eccentric J°. This conclusion renders possible the 
application of the valve diagram, as will be shown hereafter. 



91 



The Gonzenbach Cut-Off Gear. 

The valves of this gear are like those represented in Fig. 57, 
«ach of them being moved by a separate eccentric. 

To investigate the action of the cut-off valve, we must con- 
sider that the ports are fully opeued when the valve is in its 
neutral position, and that the port opening is reduced by a 
movement of the cut-off valve, no matter to which side the 
movement takes place. For the latter reason it is indifferent 
whether the cut-off eccentric 7° is secured to the shaft in the 
position shown in the diagram, Fig. 61, or diametrically oppo- 




v 

Fig. 61. 

-site. The four ported valve (Fig. 57),having the same effect as 
a valve with only one port of four times the width of the ports 
shown, and four times its travel, we should consider the ideal 
eccentric of the movement of the cut-off valve to be in J°. the 
distance OJ° being equal tofour times the throw, O 7°, of the ec- 
centric 1°. The lap-centres, Q and Q°, of the valve diagram, Fig. 
62, can be constructed in the usual manner by rotating the d'a- 
gram Fig. 61, through 90° and invertiugit, i. e , by carry ii g the 
■angles of advance, S and 6° from the horizontal centre line in 



92 

a direction oppovnqj the rotation of the crank-shaft. The cut- 
off valve is in fact a double valve (closing the ports whem 
moving either way), with a negative lap, on both sides, equal 
to the collective width of the ports, and the lap circles L° can- 
accordingly be drawn. The opening of the ports for the crank, 
angle O A. for which the collective movement of the valve, i. e*. 




Fig. 62. 

Ihe real movement of the valve multiplied by the number 
<xf ports, is represented by the line Q° k, is found by deducting: 
this movement from the collective width of the ports (collect- 
ive negative lap) Q°a, hence the distance, ka, sh^ws the 
-sought total opening of the ports. We can therefore conclude 
that the cut-off valve will admit steam as long as the crank- 
line passes through the inside of the lap circle L°. 

The diagram (F g. 62) can now be completed by adding that 
of the main valve, when the operation of the double valve will 
l>e seen very plainly. The admission of steam begins at OA+ 



93 

when the main valve opens the poit of the cylinder, the eui- >lT 
valve having been opened before, at OA°. At OE° the cut-off 
valve closes its ports very sharply, as attested by the compa- 
rative length of the line 0/j° exceeding even the length OQ or 
the maximum speed of the main valve, and expansion com- 
mences. The exhaust, indicated by the diagram of the main 
valve, is, of course, not affected by the cut-off valve. 

The most effective way of varying the rate of cut-off is a 
•change of the angular advance of the cut-off eccentric, but 
the diagram will show that tbw change is limited. 

A reduction of the angle 6° (Fig. 62) will delay the cutting- 
off, the point Q° being shifted towards Q'. This change wil^ 
•ho v\ ever, als> involve a delay of the opening of the cut-off 
valve at OA°, which should never be carried beyond the line 
OA where the main valve opens the port. 

A hastening of the cut-off can be accomplished by increasing 
the angular advance <5° (advancing the cut-off e~ccn f ric on the 
•crank-shaft) when the point Q° of the valve diagram will be 
rshifted towards Q", and the cutting- off will take p^ce earlier 
than before. This measure must, however, not be carried too far, 
;as the opening of the cut-off valve, at OA°, might be has'ened 
so much as to take place before the main valve has closed, at 
OE\ the admission at the preceding stroke, and steam would be 
admitted to the cylinder twice at every stroke. This would 
evidently be very wasteful, as the steam admitted after the 
seer nd opening would not return an adequate amount of work. 

The range of variation, admissible without trespassing these 
limits will depend on the outside lap of the main valve; for 
the lap circle L defines these limits, and hence, by giving the 
main valve ample lap, the admissible ran*e of cut-off can be 
made to cover nearly all grades desirable in practice. 

The advantages of this valve gear consist in the sharpners of 
the cutting-off and in the simplicity of it3 construction; but 
•the inconvenience of having to stop the engine for changing 



94 

the cut-off impedes its general adoption. The utilization of 
the steam is somewhat injured by the circums'ance that the 
lower chamber of the valve chest remains in communication 
"with the cylinder after the cutting off, thus increasing, as it 
were, the space of the clearance. 

Problem XI.— Construct a valve gear of the above descrip- 
tion, having a maximum follow of steam of two-thirds of t ie 
atroke and a greatest port opening of l^",when adjusted to ^"" 
of lead. The number of the cut-off ports shall be three, and 
each is to be l£" wide. Draw the diagram, determine the 
highest attainable degree of expansion, and find the amount of 
cushioning if the inside lap is equal to zero. 

We give the main valve as much lap as possible by making 
it to cut off as early as the required conditions permit, which 
is at | of the stroke. The diagram (Fig. 63) of the main valve 







95 

can thus be drawn, and will furnish the following dimensions: 
Lap=lf ", and travel— 5^". The compression is found to last 
for ^ of the stroke. The cut-off valve should now admit 
steam between the crank-lines OA and OE ; and since we 
know the radius of the lap circle L° of the cut-off valve to be 
=3§", we can find its centre by two lines drawn parallel to- 
OA and OE at a distance each of 3§", and can draw the same 
tangential to A and E. Oae third of the distance <> Q° is 
the required throw of the cut-off eccentric, which is found to 
be=I§". An advance of the cut-off eccentric for the purpose 
of hastening the cut-off, can only be carried so far that the lap 
circle L' becomes tangential to the cut off line O E' of the 
main valve, as indicated. O E° will accordingly be the crank 
angle for the earliest cut-off, and is found to correspond with 
£ of the stroke. The angularities of the eccentric rods might, 
however, admit a slight opening of the cut-off valve before the 
main valve closes, at E', and therefore should the eccentric 
never be advanced quite as far. 

This cut-off gear is frequently used on marine engines in 
connection with link motions operating the main valve. The 
change of expansion is, however, accomplished by a change of 
the travel of the cut-off valve by means of a separate link 
worked by the cut off eccentric, on a fixed fulcrum. Au in- 
vestigation will show that by this plan the range of expansion 
is very limined indeed, the centre Q° of the valve diagram 
(Fig. 62) moving in or out on the line 0Q° or on a curve* not 
much deviating from this line. By combining this cut-off 
valve diagram with the diagram of the main valve for backward 
rotation, it will appear that this cut-off valve is unserviceable 
altogether, admitting steam twice at each stroke and withhold- 
ing it at the intervening period. The cut-off valve must 

•The character of this curve will be more closely investigated in a subse- 
quent chapter, where its form is of more consequence. 



96 
therefore be thrown out of gear when the engine is backing, 
by shifting the cut-off link-block close up to the fulcrum of the 
cut-off link, to keep the valve open entirely. 

The unequality of the cut off, owing to the oscillation of the 
connecting rod, can be rectified by giving the cut-off valve 
unequal lap, i. e., by length ing or shortening the valve stem, 
according as the cut-off eccentric is located at J° (Fig. 61) or 
diametrically opposite. This correction can however be made 
to suit only one angle of the connecting rod, and it is well to 
choose the angle corresponding to ^ of the stroke of the piston, 
when the Fame correction will also cover the cutting off at § of 
the stroke, the angle of the connecting rod being the same. 

In practice, the cut off valve can easily be adjusted on the 
work. But if it is desired to know the amount of adjustment 
before band, it can be found by the diagram as indicated iD 
Fig. 64. 




Fig. 64. 



!^7 



Tm<: Meykk Cut-Off Gkak. 
The valves of this gear are substantially of the type repre- 
sented in Fig. 53 and are moved by two separate eccentrics, 
located on the shaft as indicated in Fig. 59. The cut -off valve, 
however, differs somewhat, in being made of two parts (pee 
Fig. 65) that are attached to the valve-stem by means of a right 
and a left-handed thread. The valve-stem is joined to the 
eccentric rod by a swivel, and can be rotated by a hand wheel 
or other suitable mechanism, whereby the distance of both 
cut-off plates can be adjusted, even while the engine is run- 
ning. The lap of this valve can thus be changed at will. 




Fig. 65. 

For converting the diagram of eccentrics (Fig. 59) into the 
valve diagram (Fig. 66) beginners are advised to make use of 
tracing paper to locate the points Q and Qf relating to 
the main eccentric /, and Q° and Q" relating to ideal eccentric 
J°. The relation existing between the diagrams Fig. 66 and 
Fig. 59 involves the following inferences. The distance OQ* 
is equal to one-half the relative travel, and the distance Q'Q* 
equals one-half the absolute travel of the cut-off valve; and, if 
we draw the horizontal line h through the point Q' y the angle 



98 



TiQtQ and OQ'Q° will equal the angles 6° and d' of Fig. 59, 
respectively, these angles appealing, in the valve diagram, in a 
direction opposite to that in the diagram of eccentrics. The 
line Q'Q° represents manifestly the cut-off eccentric in i'.e same 
sense as the line OQ represents the main eccentric. 







E" 
Fig. 66. 

~Ey sweeping the proper lap circles from the points Q and Q', 
the diagram for the main valve can be completed, and that of 
the cut-oft valve cm be drawn to suit the lap of the cut-off 
valve for the time being. Assuming, at first, a permanent 
negative lap, we draw t lie lap circ'es -L°, as shown, and find 
the passage v (Fig. 65) to be opened while the crank moves 
from A to E°, but as the main valve opens the port of the 
cylinder only after the crank passes the line A, the admission 
of steam will take place between A j?.n_l E°. The passage •*/ 
of the main valve will remain open between A" and E'\ and 
lienee the steam will be admitted, at the return stroke from A' 
toE". r ■■" - ' 



A change of the lap of the cut-off valve requires the draw- 
ing of other lap circles from the point Q°, and the diagr? n will 
appear as shown in Fig. 67, where four grades of expansion are 
represented, two with positive and two with negative lap. The 
lower mate of Q° is omitted, being of little interest since this 
diagram shows the action on but one side of the piston. 



A glance at this figure now shows that the range of expansion 
includes all desirable grades, provided the cut-off valve admits of 
sufficient adjustment; and also the sharpness of the cutting off 
shows a fair figure. The latter is greatest when the lap equals 
zero, and becomes less as the lap is either increased or reduced 
from this poin 

However, the margin of adjustment of the cut-off valve is 
practically limited, as will be shown in the following problem,, 
and this fact puts a limit either to the range of expansion or to- 
the rapidity of the cutting off; the latter case involves & 
reduction of the relative travel of the cut-off valve, or, m 



100 
other words, a reduction of the distance of the cut-off eccentric 
/° (Fig. 59) from the main eccentric i". 

In order that the rapidity of the catting off should suffer the 
least reduction at the change of grades, it is bet to so locate 
the cut-off eccentric that this rapidity is equal at the two 
extremest g: ades, and this is accomplished if the positive lap of 
the earliest cat- off equals the negative lap of the latest cut off 
desin d. Therefore, if it should be required to make the cut-off 
adjustable between the crank -angles A and i£,(Fig. 67) the lap- 
circle 1° would stand for both extreme gi a les,aud this circle can 
b ■ tangential to both the lines OA and OE only, if the line OQ° 
is »t right angles with OQ, and this condition requires the line 
II (Fig. 59) to form right angles with the line 01. Such a 
full range of expansion is, however, not always required, and 
hence this rule is not a general one. 

The qu s i( n may be raised if this gear will ever admit the 
steam twic3 duriDg one stroke, i.e., if the cut off valve will ever 
re-open before the main valve closes the steam port. Such an 
event would be indicated by the diagram if the rotating crank 
line would meet the line OA° (Fig. 66) before passing the line 
OB 9 and this sta'e can only exist if both the negative lap of the 
cut-off valve and the angle of advance of the cut-off eccentric 
are excessive, and can, therefore, be avoided. 

This valve gear is susceptible of quite a number of modifi- 
cations. The duty of cutting off may be imposed upoa the 
inside edges of the cut-off plates, by their being farther apart 
v hile ths upper terminals of the s^e^m passages of the main 
vjlve are cios r tog ther. The difference bet ^een ths arrange- 
ment and the one considered before consists in that the relative 
movement should be exactly the opposite to produce the same 
result. We should here remember that the diagram furnishes 
only the angular position of the eccentric, leaving a choice 
between two diametrically opposite points, between which the 
decision is to be m°.d3, as explained in a \ receding chapter. 
The same question is now at istue in relat'on to the ideal 



101 
eccentric of the relative movement of the cut-< ff valve, which, 
in this iustance, is found to require a position diametrically 
opposite to the point J°, Fig. 59, and the real eccentric should 
consequently be at i°. 

The same conclusion may be reached in a somewhat dif- 
ferent way. If we consider that the points Q and Q°, that 
were obtained directly from the inverted tracing, always mark 
the passage of the valve from positive to negative movement, it 
will be understood why the lap circles drawn from those points 
as centres always marked the closure of the admission at the fore- 
stroke. But in the last ease the cut-off is effectedby a positive move- 
ment, an.l if a result identical to that represented by Fig. C6 is 
desired, the point Q° should represent the passage from nega- 
tive to positive movement, and hence its mate, Q", is the point 
that should be obtained from the tracing, and it is the line 
Q'Q", ins: end of Q'Q°, that now represents the real cut-off 
eccentric. Of course, the line QQ° can be substituted for 
Q'Q'\ both lines being equal and paiallel. 

The described mechan'sm for changing the lap of the cut-off 
valve is not readily accessible for inspection, and being exposed 
to a high temperature is apt to wear out in a short time. This 
gave rise to the invention of other modes of changing this lap. 
One of these is shown in Fig. 68. The eccentric rod IP is joined,. 



o o 







O 


V 




W 




o ; 




:- ®"^ 


-Mz 




yo 






j «i is 


O ' 


o 





Fig. 68. 



102 
at P, to the centre of a yoke, from the ends of which the two 
C& -off plates V°,V° are operated. A raising or lowering of 
the arm G of the yoke will change the relative position of the 
cut-off plates. The objection able feature of tbi3 arrangement 
is the one-sided action of forces, the valve stems being out of 
centre with the cut-off plates. 

Another modification is known as the " Rider Cut-off. "* The 
passages of the main valve have an inclined position, converg- 
ing towards one side. The ends of the cut-off valve, whose 
seat on the main valve is cylindrical, are also inclined to con- 
form with the passages of the main valve. A partial rotation 
of the cut-off valve, by its stem, will bring its wider or nar- 
rower side over the passages, and thus change the lap. 

There are quite a number of other devices for the same 
purpose, but their descriptions would occupy too much space. 

Problem XII. — It is required to construct a Meyer valve 
gear for an engine of 20" stroke, the admission of steam to 
range between 1" and 1G" follow. The rapidity of the cutting 
off at both extreme grades is to be equal to one-half of that of 
the main valve at the moment it closes the steam-port. The 
width of the steam-poit is %", and the greatest port-opening is 
to be equal to this width. The lead is to be T L", the release is 
to take place 1" before the end of the stroke, and the cushion- 
ing is to begin as early as the case will admit. 

To solve this problem we first draw the diagram for the 
main valve with a greatest port opening=f", lead= T ^, and 
release=l" (see Fig. 69). In order to comply with the con- 
dition mentioned last, the main eccentric should have as much 
angular advance as possible, hence we make it to cut off at 16". 
Next we draw the crank angle OE° for 1" of the stroke, which 
is the earliest point of cut-off required. The centre Q° should 
be located in a line bisecting the angle E°OE, and to get its 
exact location we must consider that the line Ok Qc being the 
•Patented. 






103 
point of c^nlact of the line OE with the lap circle L) repre- 
sents the rapidity of the cutting off by the main valve, and in 
order to comply with the required condition, the point k°, 
bisectiug the distance Ok, should be the point of contact of 
the line OE, and the greatest lap-circle of the cut-off diagram. 
We find accordingly the point Q° by drawing a line from k° at 
right angles to OE. The sought throw of the cut-off eccentric 
is equal to the distance Q'Q°, and the angle formed by both 
eccentrics is equal to the angle Q°Q'0 of the diagram; (or if 
we choose to cut off with the inside edges of the cut- off plates, 
the line QQ° and the angle Q°QO are to be taken instead, and 
the eccentric located like the point i° of Fig. 59}. 

e '- 




lie 



Fig. 69. 

By the di gram we find, for the main valve, a travel of 3" 
full, a lap f" full, the inside lap is equal T y, and for the cut- 
off eccentric we obtain a throw, Q'Q°=2" full. The greatest 
laps, both positive and negative, of the cut-off valve are £f" 
full, and the relative travel of the cut-off valve over the^iain 
valve=2 OQ°=2±" full. 

Since the change of lap must be accomplished on both cut-off 



104r 

plates, their total adjustment must amount to four times the-- 
g eatest lap, which is more than %l". Attention must also be 
I lid to prevent the inner edges of the cut-off plates from 
opening the passages of the main valve before the latter closes 
the steam ports of the cylinder. The danger for this occur- 
ence is greatest when the cut-off plates are farthest apart, 
having a lap=J§", and since the main valve closes the port at 
E when the relative movement of the cut-off valve is equal to 
Q°k°={%" full, the apprehended opening can be prevented 1 y 
an overlapping of \%" plus \" for seal. Hence the width of 
each cut-off plate must be made equal to this figure (lyV'^ 
plus the width of the passage of the main valve, f plus the 
lap, J§" full=3§'' f ull, and their dimension, out to out, when 
spread apart, will be about 8|-"; and since farther the absolute 
travel of this valve is 4'' full, the valve chest is required to 
have a length, inside clear, of about 13^", allowing about \' r 
clearance on each side. 

By altering the diagram, with a view of doubling the rapidity 
of the cutting off, the valve chest will be found to require a 
clear inside length of about 2U", and the valve and the cu'- 
off plates would assume rather impracticable dimensions. 

If it is desired to know the relative position of the cut-off 
plates, i.e., their laps, for any grade, say when cutting off at 
quarter stroke, we simply draw the corresponding crank angle,. 
OE', and from Q°, tangential to this line, the lap circle, the 
radius of which is the sought lap which in this instance is pos- 
itive. If the crank angle of the given grade were beyond OQ " 
the indicated lap would, of course, be negative. 

When used in connection with a link-motion, the relative 
movement of the cut-off valve changes with the grade of the 
link. While the grade of the link is changed, the point Q° f 
Fig. 70 (cempare with Fig. G7) will move (exactly as the point. 
Q' does, because the line Q'Q° represents the cut off eccentricv 



105 
-which remains undisturbed as the link is reversed. The link 
is, however, generally used at the full stroke grades ooly, and 
the diagram, Fig. 70, shows these two grades, capital letters 
standing for the forward and small letters for the backward 
rotation. We might have considered the diagram of eccentrics 
for both grades separately, when we would have obtained the 
same diagram. Now we see that the range of the cut-off for 
the backward rotation i.9 limired, the latest cut-off being at 
OE°. This condition can be improved by advancing the cut- 
off eccentric more or les3; and the relation between the for- 
ward and backward grades will be entirely equalized if the 
angle of advance of the cut-off eccentric is made=90° when 
the points Q° and q° will be removed to Q" and q ff J vertically 
.above Q' and q' at a distance equal to the throw of the cut-off 
eccentric. It will then be necessary that the greatest negative 
lap, for the latest cut-off, should exceed the positive lap for 
the earliest cut-off, as will be seen by drawing the respective 
lap circles., 




I 
Fig. 70. 



106 
A rectification of the cut off can be accomplished in a simi- 
lar way as with the Gouzenbach valve, and this discussion need 
not be repeated again. However, with the mechanism shown 
in Fig. G8, the rectification can be made to cover more than two 
grades by making the yoke out of straight, as shown in the cut. 
The effect of this expedient is equivalent to a lengthening of the 
valve stem as the point of cut-off approaches half-stroke, and 
vise versa. 






107 



The Link Expansion Gear. 

A change of the "lap " cf the cut-off valve is not the only 
means for varying the expansion, the present valve-gear ac- 
complishing the object by a variation of both the travel and 
the angular advance. 

The main eccentric I (Fig. 71) is connected with the main- 
valve in the usual way. The cut-off eccentric imparts to the 
link F°F a rocking motion on its fulcrum F. The link block 
B is directly attached to the stem of the cut-off valve, the 
latter being as shewn in Tig. 58. A change cf the cut-off is 
effected by a raising or lowering of the fulcrum jF, by means 
of a screw or any other suitable mechanism. 

When the link is raised so much that the pin 7° is in line 
with the valve-stem, the cut-off valve will be moved directly 




Fig. 71. 

by the eccentric 1° (see Fig. 72). A lowering of the link 
implies a double change; first : the angular advance is varied 



108 
by reason of the angular change of the eccentric rod I°P° 
(Fig. 71.) and second: the travel is altered, the link-block 
being nearer to the fulcrum of the link. The angular advance 
is increased by the angle BOF =/3 / , and the travel is reduced 
in the ratio of BF to P°F. We can accordingly find the ideal 
eccentric V (Fig. 72) of the corresponding movement of the 
valve. A raising of the link will bring the ideal eccentric 
to i". 

Tor different elevations of the link the ideal eccentrics will 
again form a curve resembling the one found for the shifting- 
link, and by starting with similar assumptions this curve will 
again be a circular arc. In one point, however, there exists a 
marked difference. The curve of Fig. 72, if extended, passes 
through the centre O; for if the fulcrum F of the link 




Fig. 72. 

(Fig. 71) could be lowered to be in line with the valve-stem, 
the movement of the valve would be zero. 

This curve now relates to the absolute movement of the cut- 
off valve. To obtain the ideal eccentrics of the relative move- 
ment of the same we have to shift this curve in the direction 






109 

of, and through a distance equal to, the Hr.c JO, when w& 
obtain the arc j'o°j"<, the extension cf which passes through 
the point I' diametrically opposite the eccentric I. 

The greatest absolute travel of the cut-ofF valve is equal to- 
twice the length of the chcrd I'j"=zCi" of the locus, and if wo 
draw the line Fj° tangential, in I', to the locus, this line will 
form, with the chord l'j'\ an angle, /?, that is equal to the total 
angular change of the cut-off eccentric rod, supposing that 
the slot of the link was extended so far that the fulcrum 
of the link can be lowered to be in line with the valve-stem* 
This will be clear when we consider that the vertical inclina- 
tion of the line I'j° represents the angular advance as the 
link is lowered, while l f j" represents the same as the link is 
raised all the way. 

The valve diagram can now be obtained in the usual man- 
ner, and, assuming a negative lap, as shown in Fig. C8, the 
lap-circles can be drawn, when the diagram, Fig. 73, will givei 




Fig. 73. 

us all ttie information required. We see that the range of 
variation extends from OA to OE % but we also see that the 



110 

rapidity cf the cutting off is at the lowest figure at the mid- 
dle grades that are likely to be used the most. It is true, the 
rapidity of the cutting off for those grades can be somewhat 
-augmented, by an increase of the angle c° (compare with Fig, 
72) , whereby the locus will be changed to a position indicated 
by a dotted arc, but this change is attended by a reduction of 
the range cf the cut-off, this measure materially affecting the 
limit of the latest cut-off. 

A greater curvature cf the locus, attainable by lengthening 
the link, or shortening the cut-off eccentric rod I°F°, Fig. 70, 
or both, will favor the sharpness of the cutting off at the 
middle grades. 

In the case of the fulcrum F (Fig. 71) being on the lower 
end of the link (the engine running over) the curvature cf the 
locus will be the reverse, and the valve-gear would give poor 
satisfaction, owing to the reduced travel of the middle grades. 
For this reason the fulcrum should be on that end of the link 
t.icards which the crank is rotating, in order to effect an 
increase of the angular advance of the cut-off eccentric simul- 
taneously with a reduction of the travel of the valve, and vice 
verta. 

The proper curvature of the link may be found by setting 
both the eccentric rod I°F° (Fig. 71) and the link-blcck in 
their neutral positions, when a raising and lowering cf the lirk 
should not produce any movement of the valve. One glance 
*%ill show that the curve of the link should then be an arc 
from the centre O, provided the fulcrum F be moved in the 
same arc. It is, however, more convenient to guide the ful- 
crum in a straight line, and in this case the curve may be 
determined by means of a full sized model ; (made of stiff 
paper or any other suitable material) consisting of the eccen- 
tric rod and the link, Cne end of the eccentric rod is pinned 
to the centre O of the crank-shaft (on the drawing sheet) and 
the other end is attached to the link. A pencil point is held 



Ill 

directly over the neutral position of the link-block, and the 
link is then drawn through under it, by moving the fulcrum 
of the link in the line representing the contemplated guide. 
r Ihe pencil will thus delineate the required curve, which, 
though, will be found not to be a circular arc, and it can only 
be approximated in the practical execution. This curve will 
be more nearly circular if the fulcrum F is guided in an 
inclined line, conforming as near as possible to an arc, drawn 
from the centre O of the crank-shaft. 

The lap cf the cut-off valve is a factor, which, if not prop- 
erly determined, may have a prejudicial influence upon the 
rapidity of the cut-off alreadly limited by the small relative 
travel. The relative velocity of the cut-off valve attaining 
its highest rate when the cut-off valve is in its neutral position 
in relation to its seat, a valve with no lap will effect the cut- 
ting off at this period. Eut as the rapidity of the closure 
depend i not only on the velocity cf the valve at the moment 
of the cutting off, but also on that preceding this moment, it 
is clear that a negative lap will give better results. The time 
for the closure of the port will obviously be shortest if the 
valve has a negative lap equal to one-half the width of the 
port ; for the velocity of the valve is then greatest when the 
port is half closed. Eut it must not be lost sight of that in 
this ca^e the velocity is falling off towards the moment of 
the cutting off, and the negative lap should never be much 
greater than one-quarter of the relative travel ; for else this 
i eduction is greater than desirable. Hence the negative lap 
should not exceed one-half the width of the passage cf the 
main-valve, ncr should it be much in excess of one-half the 
shortest distance of O from the curve q'Q°q" (Tig. 73). 

Eut there is still another factor limiting the admissible 
extent of the negative lap ; for negative lap may render possi- 
ble a second opening of the ports cf the main-valve before the 
latter clozes the steam-ports of the cylinder. A brief cor. ider- 



112 
ationwill, however, show that such an occurrence will br 
prevented if the negative lap of the cut-off valve is less thai: 
the outside lap of the main-valve; for at the grade of the 
earliest cut-off the passage of the main-valve will be opened at 
the point OA°, which is found by sweeping the lap-circle from, 
the mate of the point q\ 1 his line can evidently not be in 
advance of OE unless the negative lap of the cut-off valvfr 
exceeds the outside lap of the main-valve. 

The valve is sometimes made of two plates rigidly fastened 
together by a yoke, or of a plate with a rectangular opening, 
the inside edges performing the cutting off. The cut-off 
eccentric is then placed behind, not ahead, of xhe main eccen- 
tric. The demonstration relating to Figs. 72 and 73 can be 
applied to Fig. 74, but instead of the points q'Q°q", their 
upper mates are used in Fig. 74 as centres of the lap-circles 
showing the cut-off, for a reason explained in the foregoing 



T 




Fig. 7d. 

chapter. The curve of ideal eccentrics has been determined in 
Fig. 74 by supposing tne fulcrum to be on that side of tha 



113 

link /rom which the crank is rotating, in order that an increase of 
the angular advance be attended by an increase of the travel. 
The peculiarities of the valve-gear are now considerably changed, 
-and a very early cut off is not attainable except at the expense of 
the rapidity of the cutting off of the later grades. An improve- 
ment can be effected by using either no lap at all or even positive 
lap, on the cut-off valve. For although this measure is prejudical 
to a free admission of steam, as it reduces the port opening of the 
cut-off valve for intermediate grades, the adoption of positive lap 
permits a change of the angular advance of the cut-off eccentric 
to favor this shortcoming. The most advantageous lap can easily 
be found by a few trials, as will be shown. hereafter. 

This valve-gear admits of various modifications. The fulcrum 
of the cut off link may be a fixed point, and the cut-off eccentric 
rod may be attached to the link-block, while the cut-off valve- 
stem is worked from a fixed poinc of the link ; or the eccentric rod 
may be joined to a fixed point of the link, and the valve stem 
may be worked from the link-block by a radius- rod in the manner 
of the Gooch link motion. In the former case a change of the 
grade involves both an angular change of the eccentric rod and a 
change of the absolute travel of the valve ; in the latter case a 
change of the travel only is attainable. In the first case the ideal 
eccentrics will again form a curve, which, however, will not be a 
circular arc, and should, therefore, be constructed from the angle 
of advance, and the absolute travels of the cut off valve due to 
various positions of the link-block. In the second case the ideal 
eccentrics will form a straight line, to the detriment of the sharp- 
ness of the cutting off at intermediate grades. 

Problem XIII — A valve gear of the description shown in 
Fig. 71 is to be made having arrange of cut off from ^ to ^ 
of the stroke. The greatest port-opening of the main-valve 



114 
is to be 1^", the lead {", the inside lap — 0. The cut-off 
valve is to be made in form of a frame to cut off with the 
inside edges, and its greatest absolute travel is not to exceed 
double the travel of the main valve. The cut-off eccentric- 
rod, which is 36" long, is joined to the link at a point 9" from 
the fulcrum. The virtual length of the link (the greatest 
distance of the link blcck frcm the fulcrum) is 12". The 
throws and positions of bcth eccentrics and the laps of both 
valves shall be determined, if it is required that at the grade 
of the least relative movement of the cut-off valve the time for 
closing the last half inch of the port of the main valve shall 
be as short — i. e., that the relative velocity of the cut-off valve 
\" before the closure shall be as great— as possible. 

The diagram of the main-valve, cutting off at -fa of the 
stroke and giving \\" port-opening and J" lead, is found fir t, 
as usually, and the next thing is to determine the lecu:; for 
the cut-off (compare with Fig. 74). If we first assume the slot of 
the link extended to the fulcrum, the curve will start at the 
centre Q dig. 75). Since the greatest absolute travel of the 
cut-off valve is required to equal twice that of the main-valve, 
we sweep from Q, indefinitely, the arc z with a radius equal to 
twice the distance OQ. In this arc the locus will terminate- 
We also determine the crank-angle OE° for the earliest cut- 
off, and assume, for a start, the lap of the cut-off valve equal 
to zero, in which case the point g", the intersection of the 
Hues 2 and OE°, will the be outer end of the locus. Next we 
bisect the distance q"Q perpendicularly by a right line, in 
which the centre M of the arc must lie. Thereupon we find 
the total angle of variation, /?, of the eccentric rod by making 
/0=$0=3C", (to reduced scale) and fb=12 " and then by 
copying #"Q#°=/?,we obtain Qq°, which must be a tangent, in 
Q, of the locus. The centre M can therefore be finally located 
by the line QM, forming right angles with the line Qq°, and 
the locus q"Q can be drawn. 

By drawing a line from M through O, we find the point q* 



115 
of this arc, representing the least relative travel of the cr.t- 
off valve, and sweeping, from <?', a circle of a radius of f", we 
obtain the crank angle ()E\ at which the opening of the 
steam passage is J", and Ckf represents the velocity. L fter 
assuming different laps for the cut-off valve and repeating the 
foregoing proceeding, we can select from the various attempts 




Fly. 75. 



116 

that one which gives the greatest velocity. If we assume,, 
for instance, a positive lap of -g", it is obvious that Q" (Tig. % ) 
of the new locus, must be §" from the line OE°. The new 
centre M' of the corresponding locus is found by the intersex 
tion of indefinite arcs of equal radii swept from Q (through 
iW), and from Q y . The centre Q / of the least relative travel 
of the new locus, can again be found by aline from IT' through 
O, and a circle of a radius of f '' (lap plus J") gives us the 
sought velocity, 0&", by a tangent from O, which it will be 
seen, is greater than the result of the first attempt. A further- 
increase or reduction of the lap of the cut-off valve, will be 
found to furnish less favorable results whence the assumption 
of the last attempt should be adopted. 

The angular advance and the throw of the cut-off eccentric- 
remain yet to be determined. To this end we locate the point 
p of the link where the eccentric rod is attached, and by di. 
viding the locus Q"Q in the same proportion as this point, 
divides the link, we find the point Q° representing the action 
of the valve while the eccentric rod is in line with the link, 
block. Hence the distance QQ°=& s " is the throw, and the 
angle d° the angular advance of the cut-off valve. The latter 
is negative, showing that the cut-off eccentric is located some- 
where like t° of Fig. 59. 1 he throw of the main eccentric is 
2J", the lap of the main valve=|", and that of the cut-off 
valve=f". 

In proportioning the cut-off valve we must observe that the 
plates are made long enough to prevent the outside edges from 
opening the passages of the main valve before the latter has 
closed the respective port of the cylinder. The relative move- 
ment of the cut-off valve (for the grade Q") at the moment 
the main valve closes the steam port, at jF, is equal to the 
distance of Q" from the line OE, which is very nearly equal 
to the greatest relative movement, OQ"=%l", the cut-off valve 
ever attains, hence the outer lap of the cut-off plate should; 



117 

qua! 3£", with an addition of at least J" to secure a seal. To 
keep the inside edge of the wrong plate from covering the pas- 
sage of the main valve after the latter begins to admit steam, 
we measure the relative movement of the cut-off valve for the 
; crank-ang!e of admission, OA f ', which is the distance of Q" 
from this line,=2|" about, and see that the inside edge cf the 
wrong plate, in its neutral position, will be at least this dis- 
tance from the passage ; and as the lap of the cut-off plates is 
-g". their clear distance should be at least 2". In the present 
case, however, this distance will have to be greater, for other 
reasons, and there is no fear, therefore, of one plate interfering 
with the duty of the other. 

The diagram shows very fair figures as regards the rapidity 
if the cutting-off, owing to the limited range of expansion. 
This valve-gear would, in fact, not answer when adjusted by a 
governor, since the shortest attainable admission, xiy of the 
stroke, would make the engine run away when without or with 
but a light load. m 

Supposing the engine was running over, the cut-off link 
should have the fulcrum on its lower end, since the cut-off 
plates are cutting off by their inside edges. The highest posi- 
tion of the link is found by sweeping from the proper point of 
the locus a lap circle of f " radius, tangential to the line OE, 
the position cf the centre of which determines the lowest 
point, a, of the link, which is found to be so near to the ful- 
crum that the latter should be secured to the link by abridge. 

This valve gear can be made serviceable in connection with 
a link-motion only by making the valve as in the last problem, 
of two plates cutting off with their " inside edges," since 
otherwise the cut-off* valve, negative lap notwithstanding, 
would invariabhr cut off at a very early point cf the stroke 
when the link cf the main valve is reversed, the locus assum- 
ing the position Q'Q°, in Fig. 76, while if the valve is made 



118 



as said, the locus will be in Qq°, showing that the cut-off valve- 
even with positive lap, will not cut off, at any grade, before the 
main-valve closes the steam-ports of the cylinder. Tor back- 
ward motion, then, we have to avail ourselves of the link- 
motion of the main-valve when we desire to vary the cut-off, 
and this style of valves can therefore be used to advantage- 
only on engines which are mostly running forward, as is the. 
case with marine engines. 



i*» 




Fig. 76. 

A few words on the correction of the unequalities of the 
fore and return-stroke may be added in conclusion. This 
valve-gear offers facilities for correcting the cut-off for every 
grade, as the necessary lengthening of the valve stem can be 
accomplished by a deviation of the curve of the link from 
the form denned by the foregoing discussion, and since every 
grade corresponds to a different point of the link, the precision of ' 
a correction throughout the who!c range is only dependent on 
the accuracy of workmanship. ,The finding of the proper 
curve of the link will offer no difficu'ty. if the former dis- 
cussions on the subject are duly conside el. 



119 



The Bilgram Cut-off Gear.* 
The described gears, adaptable to valves arranged as shown' 
in Fig. 58, have been shown to cnt off* rather sluggishly — 
almost throughout with less speed than the main valve alone 
would— owing either to their very nature or to the practical 
limitation of certain proportions. The present valve-gear is 
designed to overcome this difficulty. 

Both valves (see Fig. 58) are operated by one single eccen- 
tric I, Fig. 77; the main valve directly by the eccentric rod, 




Fig. 77. 

and the cut-off valve through a peculiar mechanism, consisting- 
of four members, viz : the link, the rocker, the cut-off rod and 
the adjustment lever. The link AB being jointed by the pin 
A to the eccentric rod, imparts to the rocker a rocking motion 
on its fulcrum F. The rocker is of a peculiar shape, as shown 
detached in the cut, but it virtually represents a bell-crank 
<or angular lever), having an angle BFG of about 50°, 
*Putented. 



120 

the arm GF of which is about twice as long as the arm BF. 
To t'-ie extreme end G of this rocker is jointed the cut-off rod, 
by which the cut-off valve is moved. For the purpose of 
•changing the degree of expansion, the fulcrum .Fof the rocker 
can be moved in a circular arc, being attached to the end of 
the adjustment-lever GF, whose fulcrum G is a fixed point. 

The study of this gear will consist in the investigation o* 
the movement of the cut-off valve for several positions of the 
iidjustment-lever. In every case we shall proceed from the 
neutral position of the rocker (found by transferring the 
eccentric rod to the centre of the crank-shaft) remembering 
that the movement of the mechanism will be symmetrical to 
both sides of this position. Besides, we shall as usual neglect 
.all complicating influences resulting from the angularity of 
the several members*, and besides, we shall assume the 
movement of the pin A to be strictly circular and coincident 
with the movement of the eccentric L 

.Ci- --. / 






I 

\I 



... 




ztM& 



\*c- 




Fig. 78. 

At first we move the adjustment- 1 ever until the line GF of 
the rocker assumes a vertical position (see Fig. 78), the theory 



121 
for this position being the least complicated. When the mechan*" 
ism is in operation the pin A will move in a circle, and hence the 
points B and G will move in the arcs b'b" and e'e". For the 
latter arc we shall substitute the chord to simplify the theory. 
When the crank is on its centre the pin A will occupy the 
position A ° corresponding to the position of the eccentric I, and 
since the lin.v AB represents, as it were, the eccentric rod for the 
cut-off gear, we can measure the angle of advance, S°=YAA° y 
by drawing A Fat right angles to BA. The angle OFB being 
r>0°,and AB being at right angles with FB — or approximately 
so — it is evident that the angle YAA°=6° exceeds the angle 
of advance of the eccentric, 7, by 50°. Owing to the dimensions 
of the rocker, the travel of the point C, and consequently also 
the travel of the cut-off valve, equals double the travel of 
the main valve ; and hence we can find the ideal eccentric i° 
of the movement of the cut-off valve for the considered grade 
by advancing the line O /through an angle cf 50° and doubling 
its length. 

Next we move the fulcrum F towards the right to 7 ' to 
change the grade, and denote the angular change of the rocker 
by the Greek letter /?. The corresponding angular change or 
the link AB is practically the same, and the angular advance- 
is consequently farther increased by this angle. We can 
therefore draw the line Oi v , but we have yet to find its lengths 
The movement d'd" of the pin C of the rocker is doubtless tho 
same as it was before ; but being inclined, it is only its hori- 
zontal component d'd° that is transmitted to the valve, and 
the throw Oi' of the ideal eccentric for this grade will be 
less than Oi°. The necessary reduction can be made by 
drawing the line i°i' at right angles to Oi', which will be 
understood when we consider the similarity of the triangles 
Oi'i and d'd°d". 

In moving the fulcrum F of the rocker in the opposite- 
direction we would have found the ideal eccentric *",and other 



122 

positions of the fulcrum F would furnish more points of the locus 
of ideal eccentrics The angle i°i'0 being a riglit one, it will easily 
be understood that all the ideal eccentrics will be located in a 
circle of which the line Oi° is a diameter. 

These results relate to the absolute movement of the valve and 
to find the ideal eccentrics for the relative movement, we move 
the locus in the direction of and through a distance equal to 10. 
Having thus determined the \ocuaj / j°j // of the relative movement, 
we can draw the valve diagram, Fjg. 79, in the usual manner. 




Fig. 70. 

This diagram now shows that the cut-off can be adjusted to any 
point between the crank angles A and OF, as the angular ad- 
justment of the rocker is not limited. It shows, moreover, that 
this valve gear is distinguished by the decided rapidity in cutting 
off. The closure of the steam passages is very t harp for all grades 
cutting off before the half stroke; for a later cut-off, however, this 
rapidity will get less until, when cutting off at OF, the rapidity 
of the cutting off of the cut-off valve will about equal that 
the main valve. 



123 

The rapidity of late cut-offs can be improved, if desired, by 
making the arm CF of the rocker more than double the length 
of the arm BF\ whereby, the line Ci° will be lengthened and 
consequently the locus-circle will be enlarged. This change 
entails an increase of the absolute movement of the cut-off 
valve above twice the travel of the main valve. Another 
measure consists in increasing the negative lap of the cut-off 
valve; but for reasons discussed in the foregoing chapter, it 
should never exceed the positive outside lap of the main valve. 
A reduction of the angle BFG of the rocker would likewise be 
efficient, but this reduction is attended by an increase of cer- 
tain irregularities. 

The sharpness of the cut-off will in reality be slightly less 
than indicated in the diagram, from the fact that the move- 
ment of the pin A is not circular as was assumed, but is more 
or less flattened. 

The proportioning of the mechanism requires some care ; for 
on it depends largely the proper operations and regularity of 
the cut-off. To this end we draw the rocker B FC (Fig. 80) 







NC: 



124 
with the line CF in a vertical position, and the line BF at an 
angle of 50°, and make the arm BF from, three to four times 
the throw of the eccentric, and the arm CF twice as long. 
(The given figures have been tested by a number of experi- 
ments). Then we draw the link AB at right angles to i>.Fand 
make it about f of the length of CF. The eccentric rod OF 
can next be shown in its neutral position, passing through the 
end A of the link. The cut-off rod CP° can likewise be 
shown. 

To find the length and position of the adjustment-lever G?F, 
it is necessary to make a model of thin wood or veneer, con- 
sisting of the eccentric rod, the link, the rocker and the cut-off 
rod. Next we draw the orbit of the eccentric, and on it the 
exact position of the eccentric, say for every one-sixth of the 
stroke of the piston, which may be done in the following way, 
identifying the eccentric path with the path of the crank-pin : 
We divide the diameter of the orbit passing through the initial 
position of the eccentric 1 in six equal parts, and draw the 
projection arcs of the proper radius through those points, as 
shown. The next thing to be done is to attach the model to 
the drawing by joining the parts properly together with pins 
or thumb-tacks, and fastening the ends P and F° of the rods to 
two slides representing the valves. The right end of the ec- 
centric rod may be provided with a needle point which at first 
we put in the centre O, when we set the rocker directly over 
the position shown in the drawing, and mark the relative 
position of the two slides representing the valves. Then we 
make two additional marks on one of them, at a distance equal 
to the assumed negative lap of the cut-off valve, to show the 
relative position for the cutting off on either side. Suppose 
now we desire to find the proper position of the rocker for 
cutting off at the point 1. To this end we set the needle-point 
of the eccentric rod in the point 1 of the fore-stroke, and fix 
he valves for cutting off at the proper side, when we will find 



. 



125 
that the end F of the rocker cannot be moved but in a certain 
curve. This curve we mark on the drawing by setting a 
needle-point into the rocker and making a slight scratch on 
the paper. Thereupon we attach the eccentric to the point V 
of the return stroke, fix the cut-off valve to the point of 
closure of the other passage of the main valve, and mark 
another curve by the point F of the rocker. The juncture F\ 
of these curves, must of necessity, be the exact position of the 
fulcrum of the rocker when we desire to cut off at one-sixth of 
the stroke. In this way we can find the required position of 
the fulcrum for all the other grades, which will be found to 
form a curve. By substituting a circular arc for this curve y 
osculating as closely as possible, we obtain the location and 
length of the adjustment-lever GF. An arc can generally be 
found to agree with the constructed curve with an almost 
absolute precision ; and hence it will be seen iiat this valve- 
gear will admit of a practically perfect equalization of the 
difference between fore and return stroke. 



127 
ZtsTOTIE S. 






